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.

]]>The major change is the addition of chapter 1, which is about the fundamentals of functions, with a particular focus on transformations. The only standard parent functions dealt with here are linear and absolute value functions, as this is an introduction to the principles that will be used with other functions in the following chapters.

I’ve rearranged some of the chapters in this revision, so the numbering is a little different than version 0.1.1 (that’s why I bumped up the minor version number.)

I am hoping that I will have chapter 2 written in the next few weeks, which will introduce quadratic and cubic functions.

Downloads are available here:

**Mr. Carter’s Algebra 2 Practice Book
Version 0.2.0 (June 2, 2018)**

Chapter 3: Polynomials Part A (formerly Chapter 5)

Chapter 4: Polynomials Part B (formerly Chapter 4)

These are early drafts of what is very much a work in progress. All content is subject to change.

Copyright Shaun Carter © 2018. Teachers may reproduce these documents for use in their own classroom only.

]]>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.

]]>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.

]]>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 http://www.graphfree.com/ 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!

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**7.1 Reciprocal Functions.** I prefer this title over “Inverse Variation”, as that’s too easy to confuse with inverse functions.

**7.4 Simplifying Rational Expressions****.** This also includes simplifying products and quotients of rational expressions

Downloads are available here:

**Mr. Carter’s Algebra 2 Practice Book
Version 0.1.1 (January 30, 2018)**

Chapter 7: Rational Functions and Expressions

A reminder that these are early drafts of what is very much a work in progress. All content is subject to change.

Copyright Shaun Carter © 2018. Teachers may reproduce these documents for use in their own classroom only.

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]]>At the same time, the math involved in the perimeter of polygons is pretty simple: its the sum of the lengths of the sides. There’s a reason it’s introduced at third grade (in the Oklahoma standards, at least.) But while adding a set of numbers is pretty easy for a high school student, I wanted to layer in more challenge.

My students need to figure out the missing side lengths of the polygon before they can make that simple calculation. Or, they’ll get the perimeter and have to figure out something else. Given this unit has already covered theorems related to quadrilaterals, and the previous unit was on the Pythagorean theorem, special right triangles and trigonometry, I had lots of options of for how to make students determine the information they need.

Also, students complain all the time about word problems, which tells me they probably need more exposure to and practice with word problems. So I gave my students ten problems, for which they had to draw a diagram (as I hadn’t given them one) and show all their mathematical working.

]]>These seem simple, but there are subtleties to understanding what area really is, which I’ve even seen calculus students mess up with. If you ask students what area is, they often respond with something like “area is the inside, and perimeter is the outside,” but then can’t elaborate on that. The common confusion between concepts of length and area is demonstrated by students frequently stating the area of a shape in length units.

This is why I like using the cutting and pasting of paper to represent area. I find students can conceptualize the amount of paper used to make a shape much easier than the abstract idea of “area”, even though they are fundamentally the same thing. Students can understand that one shape has more area than another using the fact that it took more paper to make. Also, if you can demonstrate that two shapes can be constructed from the exact same amount of paper, students can understand that they have equivalent areas.

This activity uses the area of a rectangle to find the areas of other polygons. Students are given a page with templates for the shapes they’re going to create, and a colored half sheet of rectangles and octagons. (The colored half sheet is actually double the number of shapes they need, but it makes things easier to have spares if they make mistakes.)

Everyone also needs a glue stick, a pair of scissors, a ruler and a pencil.

The white sheet can be cut into the six separate sections. On the back of each section, there are instructions on how to construct the given shape from a rectangle.

Students should follow the instructions to construct each shape from one of the small colored rectangles. Reactions will range from “this is easy”, to “I think I’ve got it, can I check what you did”, to “this is impossible!” Make sure you’re cycling around the room a lot, because students can and will fall behind quickly if they’re not paying attention. Frustratingly, I found most of the students who thought the instructions didn’t make sense hadn’t actually taken the time to read them step-by-step.

I suggest doing them in the order of rectangle, parallelogram, triangle, trapezoid, kite and polygon. Once they’re done, they’ll look something like this:

The idea is that each shape has an area equivalent to a rectangle in some way, because they used the same amount of paper. The triangle and the kite are related by a half, because one rectangle was able to make two of these shapes. The regular polygon is a little different, as it starts with the polygon and is deconstructed into half a rectangle; we know it’s half, as we already showed that triangles make the area a half. Add some labels to the shapes, and you get something like this:

In the trapezoid, “m” is referring to the midsegment (or median) of the trapezoid. We’ve already covered the the theorem that says the midsegment length is the mean of the base length, which is why we’re able to substitute (b_{1} + b_{2})/2 for m without any further working.

If you’d like to do this, files can be downloaded here. Included are alternate versions for non-Americans who think “trapezoid” is a weird word. Font is Matiz.

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To be clear, this is not supposed to be a textbook. The way I see it, most math textbooks aim to do three things:

- Explain mathematical concepts.
- Give “worked examples” explaining how to do different types of questions.
- Provide a bank of practice questions.

“Explaining mathematical concepts” would be better described as “guiding students to a conceptual understanding of mathematics.” That’s not something a textbook can do because *the textbook doesn’t know the students its guiding*. I believe students are best served when they construct their knowledge of math themselves, under the guidance of their teacher.

And the “worked examples” should be the job of the students, again, with guidance from the teacher. Those examples should be developed either by individual students, groups or the whole class as appropriate, with the teacher checking to make sure mistakes aren’t showing up along the way. Many of the examples in my classes end up following a slightly different path than I’d have followed, because it made the most sense to the students at the time. But a textbook’s examples are fixed, and don’t allow students to think of how they’d solve problems themselves.

That leaves us with the practice questions. This has been, in the past, how I’ve seen textbooks as being the most useful. But the textbooks I teach with (or more correctly, don’t teach with as they sit in the cabinet) don’t do a great job of that either. They are seriously outdated, and while they make some token effort of demonstrating their alignment with Oklahoma’s standards, they’re the old PASS standards, not the Oklahoma Academic Standards we’ve had for over a year now. And most of the time, it’s hard to find questions I can use, because the sequencing of the book is nothing like that of my class (probably because my sequence was developed by myself from the standards the book is not aligned to.)

That has left me looking to other resources (typically online) to find practice questions in. But that’s often frustrating too, and can sometimes result in spending hours searching for the type of practice assignment that I feel should exist, somewhere, but I just can’t find. Sometimes I find questions that assume knowledge my students don’t have yet, or assume my students don’t know something they do and don’t go deep enough. Or they prescribe a particular method to solve a question; there’s nothing more frustrating than a good assignment ruined by “use FOIL to.” Then, with my preparation time wasted, I have to create my own questions anyway.

I’ve often said, “Someone should write a book of practice problems for Algebra 2. Sort of like a textbook, but without the explanations and examples I can provide myself.” And somewhere along the way, “someone should” became “I should.”

To date, I’ve been following these objectives when writing this book:

- All work should align to the Oklahoma Academic Standards, which I’ve mostly done by ensuring my existing units and skills align to the standards, though some questions may exceed the standards.
- Avoid telling students how to solve a problem. I do break this rule sometimes (for instance, there’s a question where students are told to use completing the square, even though it isn’t needed, because practicing completing the square was the point of the question.)
- Provide scaffolding through the sequencing of questions, rather than giving students too many instructions.
- Where possible, provide some backwards, “Jeopardy”-style questions. As in, the solution was …., what was a possible equation?
- Where possible, provide “Further Practice” sections, suggesting to students how they can create their own questions (often with a partner) if they’re looking for, well, further practice.
- Look as professional as possible. I’ve learned a lot about LaTeX recently…
- All questions are originally my own. I own the copyright, so that I can use it how I like.
- I’m sure there were others, but it’s late and I really need to get to bed…

Working title is “Mr. Carter’s Algebra 2 Practice Book.” But I don’t want it to just be mine, I want other teachers to make use of it as well. To that end, I’m planning to make various draft versions available to download even at this *very* early stage.

**Mr. Carter’s Algebra 2 Practice Book
Version 0.1.0 (January 14, 2018)**

Please, have a look through it, use it in your classroom if you’d like. I’d like suggestions, though please don’t be too harsh. As I’ve said, even these chapters I’m sharing are in a very early state. I’m actually feeling nervous about sharing this, as a lot of work has gone into it, but there’s a lot more work to do. But I’m taking a risk here, because I believe this project can do what my blog also aims to do: to make me a better teacher, and possibly help other teachers out along the way.

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