Adding and Subtracting Rationals

These are the notes I wrote for adding and subtracting rational expressions last year in Algebra 2.

On the first page, I wanted to make the connection between rational expressions and fractions explicit, so I started with a reminder of how to add and subtract fractions. The two examples were chosen deliberately; the first only required changing the denominator on one fraction, while the other required changing both denominators. Students need to deal with both types of problems with rational expressions.

The next part introduced two simple problems, with only one denominator being changed between them. They are both problems that require simplifying, as I wanted to emphasize the need to do this from the start. (Students had already seen how to simplify rational expressions and stating excluded values.)

The next page was about finding the lowest common multiple. I had students use a strategy which emphasizes the definition of the LCM, having them multiply by factors so that the two expressions are the same.

And finally, a couple of examples putting all of this together.

Downloads are available here.


Which “range” are we talking about?

I’m becoming a student again this fall. As I’m preparing for that, I’ve been reading through some of the textbooks for the classes I’m taking, which has got me thinking about how we use the word “range”.

I’m talking about the range in the context of functions. The fact that I’m even having to clarify that I’m not talking about statistics, or an interval of integers, or the multitude of uses the word has outside of math, highlights just how ambiguous the word already is, without mentioning the ambiguity just within the context of functions!

To highlight the problem, I posted this poll on twitter last week:

The correct answer to this question is… well, there isn’t one. It is a (deliberately) bad question because the meaning of “range” is unclear.

The question I was really asking was “Does the word ‘range’ refer to the image, or to the codomain, of a function?” And while most said that it refers to the image (which is what I would’ve said, prior to thinking about this), 15% chose the codomain.

The problem is illustrated really well by this image from Wikipedia. The is domain is red, the codomain is blue, and image is yellow. But the range is either yellow or blue, depending on context. According to my poll, and the limited research of high school standards I’ve done, most would say yellow. But definitely not all agree with that.

That Topology book that I mentioned earlier uses the term “range” to refer to the codomain, and uses “image set” for the image. But, there is also a footnote stating,

Analysts are apt to use the word “range” to denote what we have called the “image set” of f. They avoid giving [the other set] a name.

Maybe the 20 people who all voted for ℝ were all topologists? Probably not. But, it shows that different people have different assumptions about what the range of a function means.

With all of this confusion around the meaning of the word, how are high school students supposed to understand what the range is? One solution would be to get rid of the word altogether. The words image and codomain already describe the two sets without any ambiguity, so we can just use those instead, right? We just need to define the two terms clearly, and the confusion in our classrooms will be significantly reduced.

Alas, students are still going to come across the range, whether that’s in a textbook, on a standardized test, or in their next class with another teacher. But there’s somewhere else we can turn for guidance. I’m a strong believer in the idea that whatever academic standards apply to your jurisdiction, you should follow them, regardless of your opinion of those standards. So if your standards use the word range, your class should too. I took a look at a few different standards to examine how they suggest the terminology of functions should be used.

Common Core is a bit of a disappointment here. The glossary does not define range, domain or even function. The standard CCSS.Math.Content.HSF.IF.A.1 states that a function “assigns to each element of the domain exactly one element of the range.” Which is an accurate description of a function, but it’s unclear if the range here refers to the image or the codomain.

Oklahoma (where I’ve taught most recently) defines the range in the context of relations, rather than functions: “The set of all the second elements or y-coordinates of a relation is called the range.” Now, this is not how I would’ve written this definition, but it is something we can work with. By this definition, the only values in the range are those that have a matching value in the domain, meaning the range is identical to image. If I were teaching high school classes again this year, I would teach the range as I have previously, but I’d also take the time to define the codomain as well.

The Australian Curriculum gives a nice, clear definition of the range, which matches the image, as well as defining the codomain, in the glossary for Mathematical Methods, under the heading “Function”. I don’t remember teaching the codomain explicitly when I last taught in Australia. Perhaps I should have.

Regardless of where you are, I’d argue that it’s important to define the range very clearly, according to your location’s standards. But also define codomain or image, depending on how range is defined.

The point I’m trying to make is this: As math teachers, we need to constantly examine the terminology we introduce to our students. Because sometimes we might not be being as precise as we could be. If you want your students to be really clear on the meaning of mathematical terms, make sure you are first.


Moving On

Today was our last day at Drumright High School.

Sarah and I have decided that now is right time to move on to new opportunities, with the added benefit of being closer to her family. Sarah has shared what her plans are, but I thought I should write this post to explain what I’m going to be up to.

I’ve decided that if I’m going to go to grad school, now is the time to do it. Rather than study education like Sarah did, I want to study more math. I’ll be working towards my M.S. in Applied Mathematics at the University of Tulsa. I’m really excited about being at TU, not just because of the convenient geography of being in Tulsa. Since I’ve moved here, I’ve been regularly attending the math teacher’s circle hosted by the TU math department, so I’ve already met a number of the faculty. Also, Sarah was a student there, too.

I’m not stepping away from education completely, as I’ve been awarded a teaching assistantship. I’m not entirely sure what my responsibilities will be yet, but I’m happy that I’ll get to keep teaching while furthering my own study.


Area of a Circle

Whether it’s polygons in geometry, or under a curve in calculus, I have a favorite way to explore area in class: cut up shapes made of paper and glue them back together in a new way. This time, I’m applying the idea to visually prove the formula for the area of a circle.

If it isn’t clear, the parallelogram-ish shape was originally the same as the circle at the top of the page, but its sectors have been cut apart and glued into the alternating up-and-down pattern that’s shown.

I printed eight circles to a page, and cut them into four sections so that each student could have two of them. There have been past years where I’ve had students draw and cut out their own circles to do this activity, which has the nice side effect of showing that students with different sized circles still get the same result. However, I decided this year that it was more important to get students into the activity quickly, so I gave them the template to use. Also, the sized circles I used (the radius is 1.35 in) seem ideal for fitting in a composition notebook.

You can download files for the circles here.

A few pointers on how to approach this lesson:

  • I wrote clear instructions on my board to only cut apart one circle. If you don’t emphasize this, you’re likely to have a kid who has to glue an extra circle back together unnecessarily.
  • Encourage students to get on with the task quickly. I find that while cutting and pasting shouldn’t take very long, students can drag things out if given the opportunity, and sometimes feel like they’re doing work even if the pair of scissors in their hand. Even the students who showed up five minutes early were told to get going the moment they entered the room, which set the tone for the following students.
  • Also encourage students to be precise with their gluing. Some of my kids had strange looking shapes that either curved down the page, or had large gaps between each sector and couldn’t fit the whole thing on their page.
  • Students will finish the gluing part of of their notes at very different times. I was prepared by having the notes finished in my notebook, so students could copy them as they needed. I also allowed them to take a picture of the notes on their phone to copy from, so they wouldn’t have to wait for another student to finish. Also, because students were finishing at different times, I had other work for students to go on with when they were done with the notes too.
  • I waited until most students had finished gluing before we discussed the meaning of the activity. I tried to prompt the students themselves to recognize what’s going on here so they could explain their understanding to the rest of the class; this worked to varying degrees in my different classes.

These notes include the formula for the area of a sector, but our justification of it is not included on this page. This post from a few years ago outlines how I like to introduce that concept.


Analyzing Polynomial Graphs

Here’s an INB page I created to introduce students to analyzing graphs of polynomials.

Each graph is repeated three times, so we can (literally) highlight different aspects of it. Luckily for me, Sarah is amazing at acquiring classroom supplies, so I have a lot of highlighters for students to use.

The first part was identifying the x-intercepts and the nature of each of the intercepts. I had students highlight the curve around each intercept, to emphasize whether they are simple intercepts, vertices (local minima or maxima) or inflection points.

As students did this, I tried to direct the conversation to figuring out why particular polynomials led to particular types of intercepts. This was actually really easy, as the class were asking and answering these questions without much prompting from me at all.

Next, we found the intervals for which each polynomial is positive, and for which they are negative. Having students visually represent the sections which are positive and negative really helped them in identifying those intervals.

I just (as in, while I’m writing this post) had an additional  idea to help with this part. If I’d given each student a card, they could place the edge of it along the x-axis so that only the positive parts of the graph were showing. They’d highlight those parts of the curve, then flip it over so they could highlight the negative parts of the curve.

Finally, they highlighted the sections which were increasing, and the sections which were decreasing. To find each local maximum and minimum, I just had to quickly teach them some differential calculus…

… just kidding. We used Desmos.

Joking aside, I do like using topics like this to start hinting at the math that students may be seeing in the future. I was able to explain that a big part of calculus is looking at the rate and direction of functions, with a particular focus on where functions are neither increasing or decreasing.

If you’d like these notes, downloads are available here.

I used to make the graphs. I know I’ve made my own graph sketching tool before, but it’s really only capable of parent functions and simple transformations of them, so GraphFree was exactly the tool I needed this time. (To be honest, the main reason I’m mentioning GraphFree here is I’d forgotten what GraphFree is called when trying to find GraphFree the other day, so I want to remember that GraphFree is called GraphFree. GraphFree.)

Following this, we did further practice using section 6.4 of the practice book I’m working on. Follow that link if you’d like to get those practice questions yourself – for free!