I had an initial draft version of these notes last August, but they had many problem that I’ve had to work on. The biggest issue was that I hadn’t actually taught through a Pre-Algebra course before. Covid, changing so many things like it has, meant that I couldn’t use printed notes as I’d hoped at the start of the school year, but as the year progressed and our understanding of teaching in a pandemic developed, I was able to start incorporating printed notes at times. Following the end of the school year, I went through the entire course in detail, using my students’ experiences over the year to refine the notes and even expanded certain topics that need more time to develop. The result of that review is what I’m sharing here.

**Pre-Algebra Notes Downloads (v. 0.1)**

I plan to keep improving these notes, as well as produce other material to support them. I have a first draft of practice exercises for the first three units, though those still need much more work before I’m ready share. Eventually, I’d like to create a catalog of resources and activities, aligned to each section of the notes, that would give any Pre-Algebra teachers (including myself) a huge leg-up in developing their own curricula, but that dream is still a long way off.

A note of caution that I’ve written these notes with the Oklahoma Academic Standards in mind. (Standards alignment document coming soon, hopefully.) While I’d love for as many teachers as possible to use these notes, be aware that I haven’t referred to Common Core or whatever other standards you might be using at all, so there may be some topics you expect that are missing or incomplete. I’d love to fill in those gaps for other states and countries one day, but no promises.

]]>It is Linear Function Practice: Rule to Table to Graph, and its name pretty well describes what it’s about. Students are presented with a rule for a linear function. Using that function, they construct an input-output table. Then, students plot the graph of the function using the points in the table.

The activity is self-checking. As the table is filled in, check marks indicate whether each row correctly satisfies the rule. Then, after students plot their graph, they can have the correct answer shown underneath their work. They can then go back and fix their graph if they’ve made an error. Of course, students could abuse this and look at the solution before drawing their own, so I added a couple of features discourage this. The solution button only works if the graph has a line drawn on it. And the activity reports to the student how many times they’ve clicked the solution button, hopefully provoking the pride of a few students to only attempt to check the solution once.

I should note that the activity makes no mention of (and doesn’t assume knowledge of) slope and intercepts. At this point, I’m just trying to make sure my students clearly understand the connections between the different representations of a function. At its most basic, a plot simply represents the ordered pairs that make a function as points on a coordinate plane – slope and intercepts are just characteristics of that plot that we observe. My classes have not reached the point of making those observations, yet.

Each screen is a separate question. In fact, each screen (except the last, where students choose their own function) is essentially the same. The only changes are the function and the input values that prepopulate the table. This ~~is basically because I’m lazy and wanted to copy and paste each screen~~ means that it shouldn’t been too hard to edit the activity. If you want to add or remove functions, or change them, or change the entries in the table, feel free to do so! In fact, one intention is that others be able to copy a screen into their own activities with a minimal amount of effort.

If you’re looking to customize a screen, you’ll find there’s a math input box which doesn’t actually appear to students. Look in the CL here, and you’ll find the definition of the function for the screen – the rest of the CL on the screen points back to this.

I found this worked really well with my students. It was a lot better than my original plan would have been, which was to have students practice on graph paper and send photos of their work to me. Having the activity be self-checking is essential when students are working remotely and asynchronously, as mine were this week. Students knew immediately if they were doing something wrong, and if they couldn’t figure it out on their own, knew to come video chat with me. Of course, the dashboard in Desmos is invaluable for observing how students are faring with the work from afar.

The biggest problem I ran into was students thinking they didn’t have the correct graph because of my silly mistake. I made the solution line red. While most students drew their graph in blue (Desmos’s default sketch color), some changed theirs to red or purple, making the solution practically invisible if they had the correct graph. I’ve fixed that issue now, so your students should have a much more visible black dashed line for the correct line.

This week we also did an activity of the reverse process: taking a plot, creating a table, and determining its rule. Hopefully I’ll get that polished up enough to share as well in the near future.

]]>One of my favorite ideas to use in class is the Sieve of Eratosthenes. Even if you’re not familiar with the name, there’s a chance you’ve come across it before. It’s the algorithm to find the prime numbers which are left after the multiples of earlier prime numbers have been eliminated. (Given the name of this blog, there’s not much surprise I like this topic.)

Prime numbers are not required in the Oklahoma standards for the classes I’m teaching, but I thought the Sieve would make a good start of the year activity. I’ve shared an activity and worksheet based on this, but that was from a more naïve time, when sharing papers and colored highlighters seemed like a good idea.

So I went looking for a digital alternative, and after trying a few different approaches… well, you’ve seen the title of this post. The solution, as it so often is, was Desmos.

This is what I came up with:

https://teacher.desmos.com/activitybuilder/custom/5f3eb1f440d78831973bcd4e

Weirdly enough, this is the first activity I’ve created using the Activity Builder and the Computation Layer in Desmos. But over the last few years, I’ve either been in grad school, or teaching where I didn’t have one-to-one devices.

The activity is very similar to the worksheet I shared above. It steps students through identifying prime numbers, crossing out the composite numbers as they go. But instead of literally crossing out the numbers, students type the list of numbers and Desmos takes care of the crossing. Here’s a partially complete step:

There are some limitations here. I’ve only gone to 100, rather than 150 as I did in the original worksheet, because it’s a lot easier to use a square grid in this case. I preferred 150 originally, because it means crossing out the multiples of 11 makes a difference. (I did consider using something other than 10 by 10, but the grid is already getting a little cramped at this point.) Also, typing in the lists of the multiples gets tedious. Though, I guess that’s no more so than crossing the numbers out by hand. If anyone from Desmos is reading this: would it be possible to allow tables in the Activity Builder to automatically follow a pattern, as they do in the calculator?

This is the final result:

(Note that this image is taken from this graph rather than the activity itself, but I did use the graph to make the activity.)

If you’re looking for a way to discover prime numbers in class which doesn’t involve a paper grid, hopefully you’ll consider giving this a look.

]]>There’s a lot of small fixes here, with things that could be written clearer, some additional examples, and, sadly, a lot of typos and other mistakes. The most substantial edits were to section 8.4, on natural exponents and logarithms. I’ve always been dissatisfied with the way the value of the constant *e* is justified in classes which precede calculus. I’m hoping that this explanation will satisfy students by alluding to the calculus, without teaching it explicitly.

Downloads are available here:

]]>In my last post, I made available notes for the entire year of Algebra 2, following the Oklahoma standards. I mentioned that the notes are a work in progress. Hopefully in time for the new school year for most, here is an updated version.

Changes focus on the first six chapters, which should get anyone following the notes in order through the fall semester. I aim to have version 0.3, which will focus on fixes to the remaining chapters, done by Christmas.

Changes include:

- Fixes to lots of errors and typos, both in the text and in the worked examples.
- A number of paragraphs rewritten to be clearer.
- A handful of additional examples.
- Additional space for select examples.
- Additional diagrams to explain concepts, including the picture below.

As before, there is a set of blank notes for students, and a set of the notes with all the blanks and examples completed. These notes remain free to use for teachers to distribute to their students in their own classes.

Download here:

]]>I’ve been working on a big project over summer: a set of student notes with fully worked examples for an entire year of Algebra 2.

You might be wondering why I did this. I’m not even teaching High School currently, as I’m taking a couple of years to attend grad school. While I’m doing some teaching, it’s been as a TA for Calculus 1, certainly not for Algebra 2. Even so, I’ve still spent a lot of time thinking about Algebra 2 curriculum over the past year, and had a lot of conversations about it with Sarah, as she is has been teaching Algebra 2 since last August. It turns out, even after taking some time away, my passion is still for high school math.

These are the ideas I’ve had for the function of these notes:

- They’re aiming to meet the Oklahoma Academic Standards, though in places they step back to strengthen the conceptual foundation, and in others they go beyond the standards. Eventually I’ll produce an alignment document to explain all the links.
- This is not a complete curriculum, but I see it functioning as a “skeleton” on which a complete curriculum could be built.
- The notes focus almost purely on the mathematics, not on “real world” applications (with a few exceptions in statistics topics). This is not because these are unimportant; on the contrary, they are vital. But I believe these are better addressed using methods other that pre-prepared notes.
- The intent is that the notes would be hole-punched and kept in a binder. This means if a teacher doesn’t like how I’ve done something, they can change it. Remove the parts you don’t like. If you don’t think there are enough examples, add more.
- Teachers can incorporate the notes and examples as they wish into their lessons. While textbooks give they answers to examples away from the start, with fill-in-the-blanks, the teacher can choose at what point in the class discussion they make the correct answers known.

You might remember that I was working on a book of Algebra 2 practice questions. That’s still ongoing, but it’s been overtaken somewhat by these notes. But that’s okay, because I see these as two aspects of the same long-term project. Having the notes planned out should make planning questions a lot easier.

If all of this sounds good, here’s the great news: I’m going release the notes as I work on them. And while they’re still just a first draft at this point, that first draft is entirely done, and hopefully in a usable form for the upcoming school year.

Download them here:

]]>This took a bit longer that I’d expected to create. I’ve a little experience creating graphs, so they weren’t too bad, but tables in LaTeX can be a little fiddly to get right. I also wouldn’t claim that took much thought went into choosing which functions to use. But, I’m pretty happy with the result.

You can download the files here. Included is the shuffled set of cards (pictured above), the cards sorted into the correct order, and a zip file of the original .tex files in case you want to modify them at all.

*Edit (Oct 23): There were a couple of mistakes in the original version, it has been fixed now.*

Wonders of the algebraic language

cc @cgironella2005 @mathequalslove pic.twitter.com/Bp8HfmqhDg— yair.es (@notemates) August 6, 2018

She wasn’t sure how she would do it. I tried suggesting an idea I had, but we decided it might be easier if I just made it and let Sarah see what she thought. Turns out that she loved it:

Excited to laminate this poster tomorrow and get it hung up in my classroom! Special thanks to @theshauncarter for making this poster for me after I showed him a tweet from @notemates. #mtbos #iteachmath pic.twitter.com/T5jA5LD1SW

— Sarah Carter (@mathequalslove) August 7, 2018

She did get it laminated today, and it looks like this:

What I love about this is the way it demonstrates the differences between functions, expressions and equations, but also shows how there’s a connection between them too. This poster goes well with the set Sarah created about solutions, roots, zeros and x-intercepts; that’s how she’s got them in her room, after all!

Now, the poster doesn’t capture all the particulars of these algebraic tools; it’s just a simple poster showing one example. There were some comments on twitter about the fact this poster doesn’t completely define what a function is, particularly absent a discussion of sets. Also, there was concern that the equation example implies all equations are homogeneous. Both criticisms have an element of truth, but also miss the point of a poster that, by its nature, only has one example. I think posters serve the best as either reminders of topics already discussed or starting points to launch a deeper discussion or investigation. Please, never put a poster on a wall and assume that means you’ve taught a topic. For an example of how I’ve discussed functions before, see these notes that I used in Algebra 2 last year.

This one may be a little tricky to put together, so if you want some guidance, keep reading:

- Choose your three colors. The effect works best (and helps communicate the idea) if the middle color appears to be a blend of the other two. If you don’t want to think too hard, just use blue-green-yellow like me, but I’d love to see other color combinations too.
- Print the first page (function) on the first color (blue, in my case), and the second page (equation) on the last color (yellow).
- Print
*either*page on the middle color (green). For this page, you only need the expression rectangle, which appears on both pages. Cut off the rest of the paper. - Stack your three pages with the expression rectangle of each page overlapping, with the middle color on top. Make sure you line up the rectangles as perfectly as you can.

If you haven’t heard of these before, the solution to these puzzles consist of four numbers. The top number is the product of the left and right numbers, and the bottom number is the sum of the left and right numbers. The puzzle consists of the diamond with two numbers filled in; the task is to determine the other two numbers.

If the puzzle gives the left and right numbers, the solution is pretty simple. Even so, I provided some of these on the first page so students could understand the problem before moving on to the actual challenge: finding the left and right numbers when given the top and bottom.

You may have noticed that the solutions to these puzzles are related to quadratics. For instance, the first example matches the factorization x^{2} + 9x + 14 = (x + 2)(x + 7). The magic of these puzzles is that they start teaching kids this vital skill for Algebra without it being obvious that’s what happening.

If you’re teaching middle school or even upper elementary school, I really recommend giving your students these puzzles. You can change the difficulty as you need, introducing negatives when students are ready, and decimals or fractions when you really want to up the challenge. The puzzle is a logical thinking problem which is not too difficult to understand, and ties directly in a skill needed in Algebra. As it happens, I used these in Algebra 2, because that’s what I was teaching, but I wish I’d known to use these puzzles when I used to teach 7th grade. Your students’ future Algebra teachers will love you for it.

Downloads are here, including blank versions for you to write in your own numbers.

There’s also a last-five-minutes activity I used a few times related to this. Once students know how the puzzles work, have a student invent their own puzzle and write it on the board for the rest of the class to solve. I had some (very competitive) students who were very determined to give their classmates a problem they couldn’t solve, and others who were equally determined to solve these problems.

]]>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.

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