SWBAT make sense of the greatest integer function and piecewise functions and sketch their graphs.

Functions are investigated using a game that is accessible and engaging.

20 minutes

The Function Game is an awesome way to get your *entire* class involved to do some math. It is engaging, accessible, and adaptable. The instructions for how to play this game are explained in the video of the resources section. I suggest watching that video to get a good sense of the game.

On the first day of instruction, this game is a good refresher of the types of functions that students have learned in the past. For you, it will be a good guage of what the class remembers. For example, you may find that they remember almost nothing about logarithmic functions. After each function is discovered and shared, you may want to review the type of function and what the graph would look like. I chose the functions that I expect my students to have experience with -- you can certainly change any of them that you'd like.

The fun really begins with the last two functions. Most likely, the students are thinking they get the gist of everything and then BAM! - you hit them with a piecewise function. Students will usually pick positive integers that are close to 1, so for the piecewise function they may be so confident that they know the solution - until you give them an input of 53 and the rule is different for that piece of the function. Hilarity, frustration, and deep thinking will ensue.

The last function may leave the whole class stumped. They will definitely understand the rule, but may have never heard of the function or know the notation for it. This would be a good time to ask students what the function is doing. Many will say that the function rounds down. At this time you may have to step in and explain that the function is called the greatest integer function and give the notation for it. It is also a good time to see if students can figure out why it is called the greatest integer function.

Some tips for the game:

1. Use a calculator or spreadsheet to program each function. It will cut down on your mistakes and will not give hints. If you can compute the function in your head, for example, they might conclude that is in an easy function and not consider any alternatives.

2. Choose a variety of inputs to give to students who think they know the function. Using decimals, fractions, and negatives will increase the fun level.

15 minutes

After we complete The Function Game, I want to transition to some graphing, specifically with the greatest integer function. This will also give us the opportunity to talk about domain and range.

After talking about the greatest integer function and going over the notation, I give the "Notes - Greatest Integer Function" worksheet to students to see if they could come up with the graph of the function. It might be helpful for students to use brute force by plugging in some values and plotting the points. Some students may be able to visualize the graph without plugging in any values. I would have students work on this worksheet with their table (groups of two or four) for about 15 minutes or so. Resist the urge to help them right away and let them know that they should discuss as a group and try to make some progress that way.

10 minutes

After students work with their groups for about 15 minutes, it is time to have a class discussion and give students the opportunity to synthesize the information that they have encountered (MP3, MP7, MP8). It might be a good idea to start with a student who has an incorrect graph and share it on the document camera. You can ask the class if they agree with the representation and then ask why or why not. Since it is the beginning of the year, it is a good idea to establish the norm that it is okay to make mistakes and be wrong - it creates great learning opportunities. Thank the student for sharing their work and then comment on how it allowed to class to have a worthwhile discussion about the content. Another approach would be to sketch a replication of a mistake you saw while looking at student work and then say that this is a common graph you saw and ask the class what they think of it. **Keep your poker face** - do not let on whether this representation is right or wrong!

When discussing the domain, range, and zeros of the function, it is a good opportunity to discuss different types of notation. Students should be able to work fluently between inequalities and interval notation. If interval notation if new for your students, it is a good time to introduce it!

After viewing the incorrect representation, select a student to show a correct graph or fix the one from the original student. Keep the focus on the process of questioning, analyzing, and justiftying, rather then on the correctness of the finished product.