Simplifying Radicals INB Pages (with adding and subtracting)

Our very first skill in Algebra 2 this year was this:

IN1: I can simplify radical expressions.

Even though it isn’t particularly related to the rest of the first unit (it’s called “Introductory Algebra Skills”, but it’s really functions and transformations, focusing on linear and absolute value functions), there are a number of reasons I wanted to start with this skill:

  • In the change over to the new (now one year old) Oklahoma Academic Standards, this content migrated from Algebra 2 to Algebra 1. Meaning my students (who mostly took Algebra 1 two years ago) missed out. So, starting with this is one way of bridging the gaps between the two subjects and the two sets of standards.
  • To a certain extent, this topic stands on its own, so I saw it as a way to get into our course quickly and let my kids start to get the feel of the rapid pace required in Algebra 2.
  • The skill is not particularly difficult, assuming students have a strong grasp already of concepts like radicals, exponents and prime factors. This way, I get to quickly evaluate their understanding of these concepts, and hopefully rectify anywhere they’ve gone astray.
  • Within the next few weeks, we’ll be graphing functions such as y = x² – 8. I would like for them to give me the x-intercepts as ±2√2.

simp rad expr

I know that writing out all the factors individually isn’t the fastest method, but I’m a strong believer in the idea that we shouldn’t give students the fastest method first. In this case, there are a lot of shortcuts, but I’m consciously not teaching them explicitly. I’m certainly dropping a lot of hints that they exist. But if a student really understands the principles here, they’ll be able to find those shortcuts themselves. If a student isn’t able to find those shortcuts, that tells me they still need to develop their understanding of radicals until they can.

There are two key reasons why I think this method is solid, despite its slowness. Firstly, it always works for this type of problem, no matter how many factors the number has, or how many variables we throw under the radical. And secondly, it exposes what’s happening underneath all those strange mathematical symbols. Radicals remove repeated multiplications (or “exponents” as we like to call them). I want my students to see that repeated multiplication in their mind any time they see an exponent, even if they haven’t written it out.

add sub rads

I feel like this second page could be improved a bit. I state the “rules” for adding and subtracting radicals, but there’s not much in the way of justifying why these rules exist. Our class discussion did tease this out a bit more. But I think in future I want to put more emphasis about why addition and subtraction are possible to simplify in some situations but not others.

These are my thoughts right at this moment. We can simplify when we add and subtract between things we know the relative size of. For instance, we know 3x and 2x are respectively 3 and 2 times the size of x, regardless of whatever x happens to be. So we can add them and get 5x. But we can’t add 3x and 2y, because we don’t know how x and y relate to each other. (If we happen to actually know how they relate to each other, it turns out there is a way to add them.)

Same thing with √2 and √5, or ³√2. We don’t know how they relate. We could always express them as decimals, but that’s the thing we’re trying to avoid so we can maintain an exact value (and that’s trivial with a calculator anyway, and ∴ boring). 3√5 and 2√5 are related, on the other hand, so we can add them and get 5√5.

On the inside of each of these foldables is a blank grid. These are spaces for students to write practice questions in. For this sort of thing, I usually write a few questions on the board for them to fill in here. I like to make them up on the spot, because it lets me gauge how the group is going and adjust the examples I think they need to see. (I’ll admit, this does sometimes backfire as I’m standing at the board, and my mind goes blank.) I don’t typically give them enough questions to fill the whole page, but I encourage them to copy in any other practice questions they want to keep as examples, off of worksheets or wherever else. I like to remind them that their INB is their only reference they have access to on quizzes, so this is good motivation for some of them.

Files are here

Undefined Terms INB Pages

This last week saw my Geometry classes finish off our introductory Reasoning and Logic unit, which means we’re ready to go with setting up the basic building blocks of geometry: the Undefined Terms.

undef terms

I took similar notes last year, but they were constrained to a table on half a letter sheet. This year, I redesigned it to give us a bit more space.

undef terms inside

My students questioned why we have terms that are undefined, especially after we went over the importance of good definitions in our last unit. I reminded them that definitions use existing terms to define new terms, but that can’t happen unless we have these few terms to build everything else on top of. (Is this a good explanation? Let me know if you know of a better reason to justify the undefined terms.)

Next was our first set of postulates and definitions for the year.

postulates

A change from last year is that these postulates are glued directly onto a notebook page, rather than folded inside a foldable. This is due to a decision I’ve made this year: all postulates, theorems and definitions will be visible immediately on the page they’re on. Examples, proofs and other content can be hidden in foldables. But if I’m going to expect students to look up certain parts of their notebooks frequently, I should make those parts easy to find.

Before we wrote these down, I did a little activity to get my students thinking about why these are true. I drew two points on the dry-erase board.

two points

I didn’t get any photos of this, so I’m re-enacting these on my computer.

Then I asked if any volunteers could draw a line through them. That went well enough:

and a line

I asked if they could do this wherever I put the points. The consensus was that they could. So far, so good.

Then I asked if anyone could draw a different line that also went through the two points. Some kids started saying yes, but then took that back once they realized they didn’t know how to do it. One even got as far as standing at the board with a marker, before handing it back to me and saying it couldn’t be done. My plan had worked as I’d hoped. The class had figured out Postulate 1 without me having to tell them what it was.

After we had written Postulate 1 down, I set my next challenge. Draw two lines that intersect at one point. No-one wanted to stand up to do this, because they thought it might be another trick question. But eventually, I coaxed one student to the board, who drew something like this:

two lines

Then I asked for two lines that didn’t intersect at all. I heard a student say the word “parallel”, so I handed her the marker:

parallel

Then I asked if anyone could draw two lines that intersected twice. But they knew my game by this point, and let me know it couldn’t be done. So it was time to write Postulate 2.

We didn’t get through the rest of the notes in this lesson, but I’ve written them up in my notebook anyway.

If you want to download these, you can do so here:

  • Undefined Terms
    .pub
    .pdf
  • Points Lines and Planes: Postulates and Definitions
    .pub
    .pdf

SBG Skill Lists 2017-18

Let’s cut to the chase. Here are my units with SBG skills (and alignment to the Oklahoma Academic Standards) for both Geometry and Algebra 2:

Last year was my first year teaching under the American system, with brand new standards as well. There were some parts of my courses I was happy with, but a lot of things I’m changing this year.

I’m pretty happy with how Geometry went last year. I knew pretty much from the moment I was hired that I was going to be teaching it, so I had a lot of time to think about how I was going to arrange things.

The biggest change this year is moving Trigonometry from the end of the year. I realized as I went through other units that it would have been useful to have kids knowing how to use right-triangle trig to solve problems, especially ones involving area. I’ve also scrapped the introductory unit, which was mostly some Algebra 1 review and the Pythagorean Theorem. Pythagoras is joining the trig unit now, and I’ll review things like solving linear equations as the need arises.

Algebra 2 is changing a lot. Last year, my first non-introductory unit was Quadratics. I found the classes getting bogged down, not making a huge amount of progress. Having to cover both graphing as well as factoring and all the algebraic manipulation that goes along with it got to be a bit too much for my kids. I never really felt like we’d built a solid foundational context for the rest of the course to rest on.

This year, the start of the course is going to focus on functions, transformations and inverses. For quadratics that means only dealing with vertex form, as well as showing the relationship to square roots. We’ll cover all the functions that way, before coming back to all the other algebra we missed along the way. So we’ll cover quadratics again, including all the factoring and solving stuff.

While it’s not proper spiraling, I guess it’s sort of a slow spiral.

I’m also teaching Statistics this year, but I won’t be using SBG for it. I feel like I need to learn a bit more about how a standalone stats class works before I attempt that.

Planner Calendar 2017-18

Last year I created quite and elaborate set of calendar pages for my Arc planner. I was really pleased with how they turned out and thought they looked really nice.

And then I didn’t really use them. There was much more space than I ever really needed, and having multiple levels of pages (monthly and weekly views) meant I never really knew where to put anything. I really want to be the type of teacher to make constant use of my planner. I’m hoping that I’ve found a layout that actually suits the way I approach my planning.

I like having a broad view of a lot of days, but a monthly page doesn’t really help. Because the cycles of school don’t really occur in months, they occur in weeks. But one week to a page doesn’t show enough in my personal opinion. So I created a design that shows three weeks to a page.

I usually create this sort of thing in Publisher or Word, but this one is Excel, using formulas to set all the dates.

Download: calendar2017.xlsx

All the values reference the date in cell A2. If you want to change the starting point of the calendar, change that value (type the full date, not just the day). If you want, say, two weeks to a page, make the columns wider. The font is Wellfleet, but feel free to change that if you want.

Algebra Graph Sketcher (aka Parabolator 2)

A couple of months ago I shared a tool I created for sketching graph of parabolas. I called it the Parabolator. I don’t know why. It made sense at the time.

Anyway, I wasn’t satisfied with it. It has a lot of limitations – mainly, it can only sketch parabolas. Also, the code behind it is a mess. I thought I could do a better job, and thought it’d be much more useful if it could handle other types of functions.

I spent a few days working on a replacement. Then we went to Australia for two months, and I forgot about it. Then I remembered it today.

So, here it is! Introducing the Algebra Graph Sketcher. I know it’s a much more boring name than Parabolator, but I guess my desire to be accurate won out over my desire to be silly this time.

For anyone who’s interested, I’m using Vue.js for the control interface, D3.js for the actual graph, and Lodash for… something? I guess this is why you should post about something while you still remember how you made it in the first place.

Parabolator: A quadratic sketching tool

There’s a new version now, with more functions! https://primefactorisation.com/blog/2017/08/06/algebra-graph-sketcher/

Yesterday was the last day of our school year, so it’s finally time to relax! And by relax, I mean write code.

I was thinking about the tasks I want to set for next year, and wanted to find a tool to help create sketches of graphs. Not plots of graphs: there’s already an obvious solution for that. No, I mean a bare sketch that shows only the most important points. It doesn’t need to precise, but it does need to be clear, and easy to copy.

I searched for a while, but couldn’t find anything that was really what I wanted. There are plenty of tools that can do the job, but not without a bunch of messing around first. So I decided to write my own.

Introducing: the Parabolator.

To be honest, the code behind it is kind of a mess, and it’s extremely limited, but it works. Mess around with it yourself to see what it does. Basically, it draws a parabola based on the location of the vertex and one other point. The vertex and the “second point” can be dragged wherever to set the parabola’s position, while the “third point” will position itself on the existing parabola when dragged. The axes can be moved by dragging the whole sketch. Each of the points can be toggled invisible, have labels added, and can be “locked” to the axes. When you’re done, click the download button to save your graph as a SVG file.

To be clear, this is not intended to be a learning tool, and the target audience is not students. I made this purely to help myself create graphs for assignments, and I’m sticking it online because I figure other teachers might find a use for it as well.

The use case I see for this is the rapid creation of a graph that can put into an assignment or quiz paper. It saves as a vector image, so it won’t create big ugly pixels when printed as can happen when a graph is created from a screenshot. One thing I happily discovered today is that SVG files can be directly inserted into a Word document.

I’m not going to promise it works perfectly. I’ve really only tested it with Chrome, so use that if you want the best chance of it working properly. I did also have success in Firefox, but Microsoft Edge has problems with the download feature. The most obvious drawback to the whole thing is that it only does quadratics. I do want to modify it to support other types of functions, but I’ll leave that for another day.

This is just a hobby project, so I’m not sure if I’ll spend much more time on it. That said, I do have some ideas about what I want to do (especially with adding other functions.) If you’ve got any suggestions, I’d love to hear them.

Making my Gradebook more managable

We have one week to go, which means one thing. Okay, it really means a lot of things, but I’m thinking of one thing in particular. Certain students are just realizing what I’ve been trying to tell them all year. Their grade is not high enough and they’re going to have to retake a whole bunch of quizzes before the end of the year if they want to pass.

Let’s try and make this a bit more positive. There’s also a large contingent of students trying to turn Cs into Bs, Bs into As, and even some trying to turn 99% into 100%.

Whichever way you look at it, one result is that I have to spend a lot of time with my gradebook, entering quiz scores from random times throughout semester 2, and fielding requests from students to know their grade. The software my district uses doesn’t make this the easiest thing to do. It separates each “Nine Weeks” and makes switching between them annoying, taking a few seconds of loading time and completely resetting the view I had open just before. Throw in that the same thing happens if I want to enter attendance, and that adds up to a lot of wasted time. My solution to date has been to open multiple web browser tabs with a different view in each. But that makes my browser cluttered and remembering which tab is which among all my other tabs becomes difficult. Not to mention, the address bar and tabs take up valuable space for seeing students and their grades.

I’ve come up with a solution. Google Chrome has a feature that can turn any webpage into a standalone web application, which is displayed as a separate window with a title bar and nothing else. It appears as a separate app on the taskbar (in Windows at least), which means it doesn’t get mixed up with the rest of my random web browsing. The software my district uses is Wengage, but this applies equally to any gradebook you can access through Chrome (I haven’t tried this with other browsers, but they might do something similar.)

To do this, navigate to the page you want to make an app in Chrome. Click the “three dots” button (I’m sure Google give that a proper name) and select More tools, then Add to desktop…

This will put an icon for your new app on your desktop (unsurprisingly).

icon

Double click that icon to open the app! If you want, you can find the option to “pin” the taskbar icon (keep it there even when the app is closed) by right clicking it.

app

To be honest, it really is just a chrome tab that looks a bit different, but that’s exactly what I wanted. By appearing as a separate window, it’s out of the way but easily accessible. If you want multiple windows (say, one for grades and one for attendance) hold the shift key while you click the taskbar icon. (Useful tip: this works for almost all Windows programs.)

multiple

Is this going to revolutionize the way I manage grading? No, not even slightly. But it has made something I find annoying into something slightly less annoying. Which, to be honest, is exactly what I need as a teacher sometimes.

My first year. Again.

Two weeks to go.

Do you remember what it was like to be a first year teacher? I do. I remember it really well, because I feel like I’ve lived through it again. It turns out that starting a teaching career again in a new country is really hard. But just like my actual first year, I know that pushing through it has taught me so much about education and will make me a much better teacher as a result.

I could make this a “these are the differences between Australian and American schools” type post, but I’m trying to avoid that. Suffice it to say, there are some very big differences, which have meant I’ve had to make so very big changes.

My math classes this year have been Algebra 2 and Geometry. All math teachers in Oklahoma have had to learn new standards this year, but I’ve had to learn new subjects. I mean, it’s still math, but it’s not arranged in a way I’ve ever seen before. My year 7 maths and VCE maths methods plans are not going to do me much good anymore. Even when teaching the same content I’ve taught in other classes before, I can no longer make the same assumptions about what students have been exposed to in previous years.

So I’ve basically had to start from scratch. I mean, I could just follow the textbook each lesson. That would certainly make my life a lot easier. But that would go against everything I’ve come to believe about math teaching since beginning my career. I’ve got no problem using a textbook from time to time, and some books come with very interesting questions or investigation ideas, but the textbooks I have available are terrible. They’re old and falling apart, they’re much more focused on procedure than understanding, and they don’t even align to our standards now.

That means that on top of teaching subjects for the first time, I’m also doing interactive notebooks for the first time. That means creating a whole heap of resources throughout the year. I know there are many others who have shared their foldables online, but when it’s the night before I have to teach a lesson, I usually end up making something myself. So many times I’ve found stuff this year and thought, “This is great… but it doesn’t really fit with what I’ve done already.”

You may have noticed my lack of blogging this year. I’ve found it really hard to find the time when I need that time for lesson preparation, and when I’ve had the time I’ve felt I need to stop thinking about school for a bit. I really wish I could’ve had more time for reflection during the year, rather than having to deal with the constant onslaught of “what am I doing tomorrow?”

I really do feel like this has been a year of learning everything over again. I’ve been bringing a lot more work home with me than I have in a long time. I’ve taught a lot more lessons that I would describe as awful than I have in a long time. I’ve been challenged with a group of students that are not very willing to give my lessons a chance. I’ve questioned my ability as a teacher a lot this year.

But I don’t want you to think I’m in a negative mindset about teaching here. Because all of this also describes 2010, my first year teaching in Australia. And I know that year was the necessary challenge to get through to be the teacher I am now. And I’ll look back at this year too, and see how much I developed through it.

I’m already excited about what I’m doing next year. I’ve already completely revamped my list of skills for Algebra 2, and have even begun writing problem sets for topics I think I can teach much better the second time through. I’ve slowly been evolving my classroom structures over this year, after seeing which things work for me and which things work, and I’m keen to implement more cohesive routines from day one next year. I know I’ll get to reuse a lot of the stuff I’ve spent all that time on this year. And having a foundation to build on top of will give me the chance to craft much more engaging lessons with more student creativity and problem solving.

Just like my first-second year, my second-second year will start with me as a much stronger educator.

ZERO!

I invented a new game for factoring quadratic trinomials over the summer break. After waiting to get to quadratics, I’m excited that this week I was finally able to play it with my Algebra 2 classes.

As I was planning, I was thinking about how to motivate teaching factoring. In particular, I was inspired by Dan Meyer’s thoughts, where he mentioned that locating zeroes is the key problem that factoring helps solve. I decided to find a way to make finding those zeroes the focus of how I introduced this topic.

This game, which I’m calling “ZERO!” is about evaluating expressions and finding zeroes. Students are in groups of four, and each group receives a set of 36 cards with a range of expressions on them. Most are quadratic trinomials, but there are some linear expressions, quadratic binomials and a handful of factored quadratics.

As a warm-up, I had students each choose a card, which I required to be a quadratic trinomial. I gave them a value for x, and they evaluated their expression with that value on dry erase boards. They then checked their answers with a calculator. My students are only just getting to grips with the TI-84, so I showed them how to store the value in x to evaluate the expression. Then I gave them a couple more values for x, which they also evaluated with the same card and checked with their calculator.

I asked if anyone got zero for any of the values of x, and a few students put their hands up. I revealed that this is the aim of the game – to get a card that evaluates to zero. The game works like this:

  1. Each group turns all of their cards face up so everyone can see all the expressions.
  2. Everyone chooses a card to place in front of themselves.
  3. The teacher chooses a number randomly between -5 and 6 (inclusive).
  4. Each student evaluates their expression with that number. I let them use their calculators so the game would go as quickly as possible, but I can see the benefits of having them do it by hand.
  5. If a student gets zero, they shout “ZERO!”* and turn their card face down, scoring one point in the process. If multiple students in a team get zero, they still only get one point.
  6. If a student scored, they replace their card for the next round. Other students can swap their card too, if they wish.
  7. Most points win. I went with first to ten points, before revising it to six, but a time limit isn’t a bad idea either.

As we worked through the game, I started prompting students with questions about which cards are the best ones to choose, and which cards are easiest to evaluate. I was also asking kids which numbers they needed to come up for them to get zero.

Students started realizing that the quadratics were better than linears, because they have two different zeroes – mostly. There are a few quadratic cards with only one zero. I decided against choosing any expressions that couldn’t be factored, because I didn’t want a student to be stuck with a card they couldn’t get zero from.

They also slowly realized that it was best to have different zeroes for their cards than the rest of their team (which is why I only allow one point per team each round). Four cards means eight possible zeroes, which is a better than even chance when there are twelve possible values for x. Of course, knowing what those zeroes are is easier said than done.

Well, until they know how to factor, that is. 😉

To play this game, you’ll need the following:

  • A set of cards for every four students. I printed each set on different colored paper so they wouldn’t get mixed up, and laminated them. I printed the word “ZERO!” on the back, but that’s not really necessary. Download here:

  • You’ll also need a way to choose the values of x. The easiest way would be to just own a 12 sided die numbered -5 to +6. Which I don’t. So instead, the next most sensible thing to do is write your own web app to generate the numbers. Wait, that’s not sensible at all. Oh, well. The good news, I already did that, so you can just use mine.

One bonus of having these cards is that I have practice questions ready to go. After going through factoring, I had students choose three cards each, which they factored and wrote as examples in their notebooks.

* I guess this part is optional.

Building Equations… Again

I’m teaching again! There’s so much that I can share about the start of my new job, but for now I just really want to blog about lesson ideas. So let’s do that.

In Geometry we’re going through our introductory review unit. I wanted to see what my students’ algebraic skills are, especially with solving equations. I decided to expand on an idea I used last year.

The original idea was that students could get a better understanding of the way equations work by constructing equations themselves. If students are going to be expected to “backtrack”, it makes sense that they should see how the equations go forwards in the first place.

So students choose a value to assign to a variable, then perform operations on that variable and value, step by step. They then exchange equations with each other, which they solve by finding the steps that created the equation in the first place.

My latest version has two main aspects. Firstly, it’s now an INB foldable.

And secondly, there’s a second part for creating problems with variables on both sides of the equation. This is a little more involved. I had students create two equations, starting with the same value and ending at the same result on the right-hand side of the equation. Then, they equated the left-hand sides of the equations to create the complete equation.

A big difference with these equations, however, is that solving the equation doesn’t take the student through the same steps as the person who created it. But I think that’s a good thing, as it highlights that equations like these require a different approach to solve. I hoping my students will recognize that having variables on both sides means that just backtracking won’t get to the solution.

I realizing that one of my go-to ways to structure a lesson is having students construct their own problems for other students so solve. It really helps to “pull back the curtain” and show students what’s really going on with different problems. Math seems completely opaque to so many students, particularly when they’re only taught procedural methods. Instead, let’s work on making math transparent.

Downloads: