Before Christmas, we finished one of my favorite units of the year in Algebra 1 - Relations and Functions. With regards to pacing, we are WAY behind where I would like to be, but there's not much I can do about that. Last year, my pacing was off in the other direction. We spent a full two months of the school year on data and probability because we got through the rest of the standards too quickly. I do know that this year's students have a much better handle on solving equations and inequalities than last year's students, so I'm hoping this more solid foundation will allow us to progress more quickly during the second semester.

Here are our notebook pages for Relations and Functions. There are some old favorites which I reuse every single year and some new pages as well.

Each unit begins with a table of contents divider. I've blogged about these before here. The first side contains a section titled "Top Ten Things to Remember." Students complete this either as we work through the unit or right after we finish the unit. I encourage my students to flip back through their notes and decide what the ten most important things we learned were.

The other side of the divider contains a list of our SBG skills for the unit. Students are required to record their initial score for each quiz and any updated scores as a result of retaking quizzes.

We started out the unit by refreshing our memories of the various ways to represent relations. For my Algebra 1 students, they were already familiar with ordered pairs, input/output tables, and the coordinate plane. Mapping diagrams were completely new to them. Each representation got its own flap on this foldable which is one of my favorite foldables of the year.

Now that we've reviewed relations, it's time to talk about a very specific type of relation - a function. To introduce the idea of a function, we completed a Frayer Model.

Next, we practiced writing sentences to justify whether a relation is a function or is not a function. I find that students need explicit practice regarding how to properly justify something like this.

Now that we know all about functions and non-functions, it's time for a function/not a function card sort. I blogged about this card sort and shared the file for it here.

Since I started teaching, I have struggled to teach independent and dependent variables in a way I am proud of. Each year, it seems like I try something new. And, no matter what, the same 60% of kids who intuitively understand independent vs. dependent get it. And, the same 40% of kids who mix up dependent and independent EVERY SINGLE TIME still mix them up every single time. This happens no matter how well I feel like I've explained it.

Here is this year's attempt:

I'm especially proud of the inside. I think that having students sketch a graph has helped their understanding of independent vs. dependent variables.

This approach, however, was not the magic cure. I still had a fair number of students who switched the variables every single time.

Now, let's take a closer look at discrete vs. continuous functions.

This discrete vs. continuous card sort still makes me so happy. I blogged about this activity here last month.

Once again, I chose to pull out the good ol' DIXI ROYD mnemonic device for Domain and Range.

I was feeling really uninspired when it came time to type up domain/range practice problems for our notebooks. Luckily, Math by the Mountain came to the rescue! These next two booklet foldables are her work. You can learn more about her relations and functions unit in this blog post.

Domain and Range of Discrete Relations:

Domain and Range of Continuous Relations:

Up next: Domain and Range Restrictions. I simply edited my booklet foldable a bit from last year to update some of the examples.

I added a new graphic organizer this year for rate of change. This resulted from making the decision to NOT introduce the term "slope" during our relations and functions unit. Instead, I decided to wait until our linear graphs and inequalities unit to begin referring to rate of change as slope.

Rate of Change Practice. We did four problems and stapled them together so they only took up one page in our notebooks.

After looking at graphs with a constant rate of change, we shifted to graphs that have various rates of change. I'm a bit disappointed that both graphs I chose were distance-time graphs. This is definitely an area of improvement for next year!

It's graphing story time! This was my second time using popcorn graphs with my students. The conversations were awesome this year as well!

I was a bit short on time, so I just ended up using the same graphing stories foldable as last year.

Here are close-ups of the two tasks inside. I found both of these tasks online.

I created this puzzle to motivate my introduction of function notation. I really liked the concept, but my presentation could use some work. All of the values I picked as my examples could be found on the same linear graph (despite the entire graph not being linear). So many of my students thought that the function notation just meant to multiply by 6. :( It did lead to some interesting conversations when some students had interpreted the puzzle as multiply by six and others had interpreted it as look at the ordered pairs that the graph goes through.

Another big change I made this year was introducing evaluating functions by only looking at graphs and tables FIRST. Once my students were comfortable with evaluating this way, then I introduced evaluating from an equation. In the past, I did it backwards of that and started with equations. I think it was too much too soon. This year seemed to work a million times better!

Evaluating Functions From a Graph:

Evaluating Functions From a Table:

Evaluating Functions From an Equation:

Now, it's time to practice writing functions and using them to solve problems. My students found these problems to be a bit difficult. I think we should have spent more time practicing these than we did.

For graphing functions, I chose to do the Win Some Cash task again from last year. Again, my students got super sucked into the scenario.

While doing the task this year, I realized a mistake I had made last year. :( Last year, I had my students connect the dots to more easily see the shapes made by exponential, linear, and quadratic functions. This year, I highly emphasized discrete vs. continuous graphs and when it makes sense to connect the dots and when it doesn't. This is actually a discrete function, so we couldn't connect the dots this year. With this in mind, I'd like to edit the activity a bit next year to bring out the general shapes of different types of functions more clearly.

Also, note to self: make the graph bigger. I made a note about that last year, and I forgot to fix it before printing it again for this year!

We lots of graphing practice. Students had to graph each function by making an input/output table and classify each function as linear, quadratic, absolute value, or exponential. My students really struggled with knowing how to connect the dots. When I instructed them to connect the dots from left to right, this just confused them even more. Not sure how to remedy this for next year. Maybe I should make some sort of connect the dots puzzle for them to work out that goes left to right...

This year, I meant to make a summary page that discusses key points about each type of function, but we ran out of time. It's definitely on the list of things to fix for next year.

This two page spread in our notebooks just makes me smile.

A Frayer Model for "Linear." Students had to create their own examples and non-examples.

Our last notebook page of the unit was a linear/non-linear card sort.

You can download the files for this unit here.

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Sarah,

ReplyDeleteThank you for sharing all your wonderful ideas! I was not able to find the locate your updated DIXI ROYD organizer or your versions of the Domain and Range of Discrete & Continuous Relations Foldables. Could you please point me in the right direction?

Thank you!!

This will save me for our upcoming functions unit! Thank you! I have one question: for the input/output table review, I couldn't find any discernible rule to the values in the table. Isn't there supposed to be a rule, or does it not matter in this case? I'm new to teaching pre-algebra, and want to make sure I give my students correct information. Thank you again for sharing your fantastic materials!

ReplyDelete-Lesley

Hey Sarah!

ReplyDeleteThanks for sharing your functions unit! It has been a huge help! I couldn't find the Domain and Range of Continuous foldables as well. Did I overlook it somewhere?

-Hannah

Would it be possible to get the new DIXY ROYD graphic organizer? It's not with the rest of the files. Thanks!!!

ReplyDeleteSarah, I love all of your information on how you teach relations and functions. I've never liked how I have tried in the past to teach this topic. I downloaded the majority of your information for this section, but I can't find the evaluate from an equation file that you show in your post. Is there anywhere else you might have it posted?

ReplyDeleteSarah, I love all your creations and activities. I find myself looking first to your blog when I am in need of something new. I downloaded your entire unit on functions and relations but am having trouble locating the DIXI ROYD and the foldables on Domain and Range. Could you please send these to me? mitchellj@benjaminlogan.org

ReplyDeleteHi Sarah! Thank you so much for having such a great blog and resources. I've been stalking it for several years now. :) Is there anyway I could get the file for independent and dependent variables? It's not included in the downloads but it is great! My email address in emily.bennett@dragons.k12.ar.us

ReplyDeleteHi Sarah! First, thank you for everything you do and post. I love your materials! Second, I did the Relations Telephone Activity and the students loved it! Multiple students said how fun it was. So thank you again!

ReplyDelete