Week 1 Teaching

My year at school has begun, and as of today, we have had a full 5 days' worth of classes, even though lots of kids were missing class here and there for special retreat-type of activities. I feel quite settled, and I am starting to learn most of the kids' names despite having a terrible memory.


I am thrilled about how Precalculus is going! It's almost too good to be true -- the kids can build their own patterns from colored blocks; they can write recursive equations to represent those patterns; they can explain the limitations of knowing only the recursive formulas; they can write explicit equations that turn out to be linear; they can sum arithmetic elements and explain why that sum must be quadratic, both from a graphical standpoint and an algebraic standpoint. And they're ready to break down quadratic patterns into linear elements, in order to find the formula for the n-th quadratic element! ...Rock ON, 11th-graders!! They will start their first project very soon, either tomorrow (if they understand all preceding concepts) or next week. It is a project that I adopted from the IB, watered down quite a bit simply because I don't want the kids to be scared off right away.

My vision of tying linearity and quadratics neatly together for the kids with the concept of sequences is working out well so far; we'll see how they fare on the first quiz! :)

In Calculus, I discovered on Day 1 that the kids were strangers to the powers of the graphing calculator. I spent the next day walking them through basic features, and since then they've been flying through the conceptual material which I've presented through explorations. Unfortunately, this class has had the most amount of absences, since the Seniors and various special leadership folks had to miss class starting on Tuesday of this week. That has caused the pacing to be a bit chaotic. The kids who have been here everyday now have a solid grasp of the derivative: what it is, how to read it from the calculator, and how it relates to average rate. They can visualize derivative graphs when looking at an abstract original graph of f, and in general I think their calculator skills are slowly maturing. Yay!

My Algebra 2 classes are two diverse groups. One is full of international students, some of who had just stepped off the plane to arrive in America with very little English comprehension. This has made for a very interesting challenge, trying to teach those kids in the same class as the native speakers. I think it is great -- it helps to teach all kids the virtues of patience and kindness, because some of the non-native speakers are much stronger in math than some of their native-speaking peers, so in my mind at least, it's a bit of give-and-take for everyone. The other Algebra 2 class is full of bubbly tenth-graders, and most of them are my advisees. I simply love that class! They don't work as fast as the other group through the material, but they're very responsible, eager to learn, and nice. Kids from my Grade 10 class keep saying that that seems like the fastest-passing period, and they seem to genuinely enjoy their time in class even though we're still on the nitty gritties of basic algebra. We're working our way through some knowledge about how to use the graphing calculator flexibly to check our work alongside reviewing skills from Algebra 1 and wrapping our mind around visual linear patterns. The first big project will be coming up soon though (next week!), and it's the rubber chicken bungee jumping project (I think at most schools, they use barbies), which will involve a significant amount of writing and really show kids what I expect this year from their analyses. Whee! I'm excited!

How are your school years going??

Precalc Breakdown

Continuing with my Precalc brainstorm, I went through my new school's textbook (Functions Modeling Change, Fourth Edition, from John Wiley & Sons, Inc.), and pulled out specific content strands to correspond to each of the core topics I will need to teach. The current Precalc teachers don't all teach sequences as a topic in Precalc, but when I took it out, I found that no matter how I sequenced the basic algebra topics, they always seemed somewhat random and disconnected. So, I will still start off with a sequences discussion, in order to tie everything together for my own students. I also brainstormed a project for each Precalc unit during the year. If I end up throwing them out, it's ok! But, hopefully this gives me ideas to fall back on throughout the year. If they don't have references, either I have that project lying around from my own creation before or I think it'll be fairly straight-forward to pull one together this year.

Thoughts??

Topic	Key Lessons	Project or Lab
From Arithmetic Sequences to Linear and Quadratic Functions	1.       Sum of arithmetic sequences 2.       Quadratic patterns and quadratic formulas from applying sequence formulas to sum 2nd differences Daily openers: Practice linear and quadratic skills	Triangular Numbers and Stellar Numbers (past IB Portfolio topic)
From Geometric Sequences to Exponential Functions	1.       Sum of geometric sequences 2.       Exponential formulas from applying sequence formulas to geometric sequences 3.       Recursive vs. explicit formulas Daily openers: Practice basic exponent and log skills	Retirement Project / writing letter to parents
Modeling and Visualizing Graphs of Basic Forms	1.       Interpretation of parameters within a formula, domain, range 2.       Compare and contrast linear vs. exponential word problem setups 3.       Using GCD to analyze word problems after proper setup Daily openers: Practice graph prediction from formula	Regression lab via RC-circuit
Transformations of Families of Functions	1.       Review function notation 2.       Use the studied functions to reinforce basic transformations knowledge Daily openers: Practice transformations, forwards and backwards to/from formulas	@Samjshah’s family of transformed functions art project
Mini Capstone Unit: Modeling with Functions	1.       Go over function forms and uses (including forms not yet studied in the course) 2.       Give students guidelines on modeling assignment 3.       Do one example during class and then provide time in class to complete individual assignments Daily openers: Adjust as perceiving student needs.	Use old IB portfolio topics
Trigonometry of Circles and Waves	1.       Review non-right triangle trig with project from @KFouss 2.       Motivate circular trig using rollercoaster problem 3.       Unit Circle 4.       Graphing and transforming waves in both modes 5.       Solving trig equations (graphical and algebraic) 6.       Trigonometric identities 7.       Tangent function and GCD solutions Daily openers: Unit circle memorization; from hand-drawn graphs to finding trig formulas (and thereafter, to finding GCD solutions for intersection points)	Non-right triangle trig project from @KFouss Building an animated rollercoaster in Geogebra
End Behaviors of Polynomial and Rational Functions	1.       Start with modeling problems to motivate the various forms of new functions 2.       Fully develop each function type completely, paying attention to Calculus terminology 3.       Graphical analysis of instantaneous rates along a graph Daily openers: Predictions of graphs based on formulas; verify with GCD	Stocks modeling project Field trip planning project / write proposal to principal
Logarithmic Functions	1.       Function inverses 2.       Motivate topic using exponential vs. log scales 3.       Mixed practice in interpreting log scale graphs Daily openers: Solving logarithm equations	Research/poster representing an exponential dataset using multiple forms of graphs, discussing visual tradeoffs
Non-functions and Modeling Using Different Coordinate Systems	1.       Circles and Ellipses 2.       Polar coordinate system 3.       Hyperbola and the idea of Locus Daily openers: From Cartesian to Polar Coordinates, vice versa	Art project via Desmos
Year-long, Ongoing Review		Create a Precalculus Magazine summarizing each unit that we have learned.

Precalculus Retirement Project

I came up with a pretty useful retirement project idea for my Precalculus class. I say this is useful because more and more, I find that a lot of our friends are stressing over what's going to happen when our parents retire. Are they going to have enough money to get by? For how long? It seems to be a somewhat complex problem to predict accurately, because each month both the interest compounds and the principal left in the bank decreases.

So, to that effect, I plan to do a retirement project at the end of studying sequences and recursive/explicit formulas, and the kids are going to write a letter to their parents to make recommendations on why planning for a continued source of income post-retirement is really essential. Hopefully, this will help their parents talk to them a bit about money and financial planning, which I find that in some families is not as open a topic as it ought to be.

Check it out: http://bit.ly/retirementProj

It's still early in my planning of the course, and I would love any suggestions you may have! Geoff recommended that I could teach the kids to cross-check the explicit formula results from their calculator tables against the results from Excel (which are essentially iterations of the recursion from row to row, when you drag the formula downwards).

Precalculus Brainstorm

I was brainstorming for my Precalc class this morning, and I realized that my perception of the Precalc topics has changed since I began teaching IB two years ago! The IB equivalent of this course heavily emphasizes the interconnectedness in applying them, which has in turn changed the way I wish to organize my Precalc class next year. I'd like to organize my Precalc class this year as a sequence of topics that each arises naturally from the previous topic, with the entire course anchored upon modeling as the core skill and purpose for building further functional knowledge.

See flow chart below. (The bold parts are what I think are the more important concepts from the course.) If my new colleagues would support my decision in organizing my course this way, then I hope to start with sequences as a way of re-introducing linear, quadratic, and exponential forms. The kids will see, for example, that if you use summation formulas to capture the sum of second differences, then the resulting n-th element will have a quadratic form in terms of n. At the end of those basic functions re-introduction, my hope is that the kids can do a written project analyzing triangular and stellar numbers, similar to the old IB portfolio task from a few years back. (I've misplaced that prompt now, so I'll have to create one that is similar.)

Then, using their knowledge of these basic functional forms as a basis, we will examine the graphs formed by these basic forms and use that to re-introduce the core concept of transformations. We will learn the other functional types only as necessitated by modeling of different types of data, so that the kids can always remember the importance of contextual analysis and interpretation. Eventually, at the end of the course, each student will do two modeling projects: 

1. a project using GeoGebra or Desmos in order to create a picture with functions and to practice basic functions modeling and specifying domain restrictions. 

2. a real-world modeling project of their choice, in order to practice asymptotic analysis and written communication. In this final project, we can add additional requirements such as analyzing the rates of change, in order to preview some introductory concepts from Calculus.

Thoughts? Do you think this organization would make sense to students?

Visualizing Complex Operations

I frequently feel sad that complex numbers are not part of the IB SL curriculum. During the time when I taught Algebra 2, it was always my favorite topic to connect algebra, geometry, and the history of numbers all at once.

Here is one cute activity I used to give to my students to illustrate the relationship between complex number operations and coordinate transformations. I recently gave it off-handedly to a student at our school, who was very intrigued by this, so I thought I'd share with those of you who still teach this topic. I think this activity is very eye-opening for the students and very visual, and it gives the various algebra operations a more concrete meaning / some motivation.

I vaguely remember that I had written about this a long time ago, but here I am posting it again since I cannot find the old post. (oops. Lost in the WWW, I guess.)

Completing the Square Geometrically

I am so excited! I taught completing the square today in my "low" grade 9 class for the first time, and I decided to try a geometric approach this time as inspired by my friend from PCMI (the awesome Danielle Soderberg*), and the kids loved it!! After just two examples on the board, they had no trouble at all working through cases with integer leading coefficients and integer vertices. It was fantastic.

Here was the worksheet I made for completing the square, and here was the warmup sheet I used prior to introducing complete the square (cut into half-sheet strips; the kids skipped the conversion to vertex form at the beginning of class and came back to that column at the end, after the lesson).

Enjoy!

*Although, as a small disclaimer, I never saw Danielle's worksheets, so it's quite possible her approach is different and/or it is far superior. She vaguely mentioned something about algebra tiles and completing the square.

Pancakes! (and Function Compositions)

I was reviewing composition of functions today with some of my students as part of a bigger review on function basics. The example question I used, a simple one, specified that
f(x) = 2x - 1 and g(x) = sqrt(x) and asked them to write the equations for (f ○ g)(x), (f ○ f)(x), and (g ○ f)(x).

I showed them how to re-write the composition notation like this:

(f ○ g)(x) = f(g(x))

and then I asked, "Is this composition primarily an f function, or primarily a g function? If you could figure that out, then writing the equation will be really easy..."

The kids stared at me in silence.

I tried hinting at it, "Let's think about which function occurs first and which occurs last."

"G occurs first, followed by f taking in the output of g as its input," so said the kids. But they still weren't able to state what the primary function is. The hint was too esoteric.

I tried again with a silly analogy: "OK, let's say you first make someone really fat, and then you smoosh them down and flatten them out like a pancake, then are they primarily fat or primarily a pancake??"

Kids giggled and mumbled more or less in unison, "Primarily a pancake!"

"OK then. If g occurs first and then followed by f, what's your primary function here? Which one is the pancake?"

Kids mumbled, "F is the pancake."

"Are you sure? Which is the primary function here?"

"F," they said. They're still not sure how this helps them write the formula.

So, together we copied down the equation for f, since we know that this is going to be the basic form/framework for our composed function:

f(x) = 2x - 1

But since we know that f is no longer just taking in a simple x value but taking in the entire function output from g, we can replace the bold parts above with g(x) and its associated formula as shown below:

f(g(x)) = 2*sqrt(x) - 1

Easy breezy. (On the board, by the way, I made sure to circle the bold parts and to draw arrows between the equations to indicate which x's got replaced with what.) The whole thing took less than 5 minutes to teach and all the kids said they now understand both how to do it and the concept behind it. (I gave them two more questions to try, just to be sure. They had to do
(f ○ f)(x) and (g ○ f)(x) on their own, and they did them both quickly, correctly, and without any doubts.)

I think review sessions for small groups of struggling students are helping me to become a more efficient explainer altogether. I find myself thinking of different ways of explaining things than how I normally structure it in class.

So yeah, pancakes! They are the magical ingredient to teaching function compositions, apparently.

Beyond the Algebra of Composition Functions

Now that my kids are comfortable with writing composition function equations, I gave them a few word problems to illustrate:

1. why composition functions can combine functions that have different types of domains. (For example, function f takes in dollars as domain, and function g takes in time as domain. It's possible to get an equation that represents f(g(x)). See first problem in Part 2 of the worksheet.)

2. how composition functions combine step-wise dependencies to represent them all in one swoop!

(The examples are a bit silly. But, they are intuitive and easy enough for kids to grasp/follow. After this, I made them do a bit more heavy-duty problems in the textbook, that are less light-hearted and a bit more "real", but also less fun.)

Check them out! (Part 1 is adopted from a lesson from NCTM. I just re-formatted the questions and re-worded them quickly. I was a bit scared by how long it took my 11th-graders to get through that first exercise.) I think the worksheets were pretty effective. I'm sure if you did it with a more accelerated class, they'd breeze right through this, and it'd help solidify their conceptual view of compositions.




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I also have an idea for how to teach inverse functions this year using a sort of telephone game. I haven't tried it out yet (that's for tomorrow's class), but I'm thinking of starting the class with writing a table of first / last names on the board and asking kids to evaluate
f(Isabella) or f(Alvaro) for last names. Then, I'll introduce / review that the reverse lookup notation is
f^-1(Alvarez) or f^-1(Garcia) and that this is called the inverse function.

Then, I'll call a few kids to sit in front of class in a row, and they each will get to pick a secret basic operation +, -, x, with an operand. For example, "+ 3" or "- 8" could be what they secretly choose. I'll give the person on one end of the row a number, and he or she would do the operation in their head, and then tell the result to the next person. So on, until they get to the end. Say that the original number is 10 and the final number is 61, I'd write f(10) = 61 on the board. Then, I'll ask them to go backwards, starting with the last person with the number 61. Each step, they should "un-do" their own step by doing the opposite of what they did before. That way, by the time they get to the front again, we should see that f^-1(61) = 10 happened by reversing each operation AND reversing the order of each operation. And I'll use that to introduce how to write equations of inverse functions!

Example: First person secretly chooses "add 3", second person "times 2", third person "minus 9", fourth person chooses to square.
f(x) = (2(x + 3) - 9)^2

That means that on the way back, in order to use the output to find the original input, we would need to: First square root, then "add 9", then "divide by 2", then "subtract 3".
f^-1(x) = (sqrt(x) + 9)/2 - 3

I think the nice thing about this demo is that we can repeat this process quickly if kids have questions about any part of the classwork/homework. (I'd just choose kids to represent each step of the operation, and have them sit in a row to demonstrate how the operations reverse their nature as well as their order.)

...I'm excited about this!! I think it will work and make sense to the kids!!! (Of course, I'll also teach them the "short cut" of flipping x and y and solving for the other variable. But I think that comes later, once they have a foundation of what inverse operations are and why they work.)

The other nice thing about the chairs exercise is that we can use it down the road to illustrate why f^-1(f(x)) = x, by putting twice as many chairs in a row, with the middle two operations canceling each other out, and then the next pair canceling each other out, etc. Example:

x --> Add 3, Times 4, Minus 1, Plus 1, Divide by 4, Subtract 3 --> you get x back, obviously!

Thoughts??

Addendum: The telephone game worked fabulously! It also was a good anchor for me to come back to in order to explain to kids why inverse function has nothing to do with flipping the signs. All I had to do was say to kids, "When I gave them the reverse input while going backwards, I didn't flip its sign, did I?"

Using the first and last names as warmup also had the added benefit of a giggle factor, when I explained to the class that f(Cuellar) = undefined. (Cuellar is the last name of a very silly and likeable kid in the class, but since it's the wrong type of domain value, it cannot be evaluated. The class thought it was funny that his evaluation gives an error.)

Visualizing Operations on Functions

For my Precalc kids, I started toying with this idea of presenting function operations using diagrams to help kids visually organize domain changes and to see how equations relate to one another.

Here's stab #1 (we started doing this earlier this week. I plan on finishing this tomorrow and easing our way into composition of functions...). My reasoning for organizing it like this is to show kids that addition, subtraction, and multiplication are all very forgiving operations. As long as f(x) is a valid value and g(x) is also a valid value, you can add/subtract/multiply them with no problem. Only division might cause additional exceptions in the domain.




I'm sort of envisioning the kids to then go into something like this, where they can picture composition as one function machine feeding into another one and using that idea to write equations. I also hope that they'll figure out on their own by the end of this second worksheet that g(f(x)) will only be undefined if either f(x) is undefined or if f(x) is some value that will in turn "break" or cause an error in g.





Finally, I'll squeeze in a game/activity, where they get in groups of 3 and split a deck of "cards." Every round I'll call out some order of composition -- for example, g(f(h(x))) -- and the kids would need to write the resulting equation and find the domain restrictions on that formula. And then, with those same 3 cards I'd reverse the composition order, and they'd do it again. It's not terribly fun of a "game", I guess, but as far as activities go, I hope it will be a little better than doing straight problems on paper, because they'd hopefully notice that the domain restrictions only come into play for certain types of operations (ie. square root), regardless whether that operation happens first or last in the composition.


Thoughts or suggestions?

...By the way, I recently read on someone else's teaching blog that they have all these really wonderful things planned for their Precalc kids. It made me feel a little math-envy, but alas, my kids need all of the basic reinforcements that they can get. (They're coming along, and are actually understanding words that are written on paper with numbers inserted in between!!!! It seems like half a miracle considering the zombie-esque state that they were in when I got them back in August.) But, our sloooowness in progress is making me very worried about their future, and I'm going to try my best not to become a total stressball over this during the next few months. It's going to take me another couple of weeks to just get through all of the Algebra 2-ish review-ish stuff with them, and then whatever trig I can squeeze in to the rest of the year, I'll have to be happy with!! The end of the year is coming SO FAST.

Function Transformations Nitty Gritties

I've been racking my brain about how I am going to break down to my Precalc kids the procedures for how to analyze the various transformations involved in a function that looks complicated like g(x) = -3(-0.5x - 4)^3 + 7... Transformations as in, when you compare g(x) against its parent function, f(x) = x^3, what types of shifts and scaling and reflections had resulted in this new function g. This topic is always really difficult for kids, because I am sure that when a regular kid looks at the many numbers inside a complicated transformed equation, all of the number just melt together and seem to be indistinguishable.

Well, I think I've got the big-picture concept finally nailed down to one picture. I hope this is children-safe. It's sort of like an input/output diagram of a function box, but annotated.


So, basically, the way I see it, some transformations happen outside of the parentheses (such as multiplying by -3 and adding 7, in my example function g...see below for color-coding) because -- if you consider PEMDAS -- they occur AFTER the "main" function of cubing has already occurred. At that point, it's too late to be affecting x-values, so you're naturally affecting only y-values and causing vertical changes. And since kids already know that higher coefficient = steeper graph, it shouldn't be difficult for them to figure out that this means -3 does the vertical flip / vertical stretch by scale of 3, and 7 does the vertical shift.


On the contrary, (if you again consider PEMDAS,) transformations that occur inside the parentheses had occurred BEFORE the "main" function of cubing occurred. So that means they were operating only on the x-values and therefore only resulted in horizontal changes. Again, most kids can figure out that a fractional coefficient of 0.5 makes the graph flatter, so it shouldn't be hard to make the leap that 0.5 = 1/2 stretches it horizontally by a factor of 2. (Since horizontal stretching makes a graph flatter, whereas a horizontal compression would have made the graph look skinnier/steeper, and therefore would have had a coefficient greater than 1 on the inside.) And -- keeping with the theme of "inside" operations affecting only the x-values -- the negative sign on the -0.5 necessarily makes the graph flip horizontally across the y-axis.

The only complication that remains to be explained is why the horizontal shift is 8 to the left, rather than something immediately visible inside the equation, such as 4 units. The way I've always thought about it, special things happen at f(0) for most parent functions. In order for you to capture that same special point on the transformed graph, you have to find the special value x that would result in you still evaluating zero in the "main" function (in this case, by the time you reach the actual cubing operation, the stuff inside the cube would need to be zero in order for you to observe the same special behavior). So, by setting -0.5x - 4 = 0 we get x = -8, or our special point (and every other point from the old graph) has apparently shifted 8 units to the left!

...Obviously, this reliance on order-of-operations can be translated (harhar) to other parent functions as well. Here are some examples, with color coding to show which operations come before and after the "main" function operation.



(Of course, my juniors will probably have trouble with simply articulating PEMDAS in abstract algebraic form. Whenever that happens, I make them actually plug a value into x, in order for them to write down all the steps of what operation happened, in which order, before they arrived at the result.)

What do you think? Is this explanation child-proof, or still way too complicated? I don't want to be giving the kids a bunch of blind rules to memorize that they don't understand, especially because there are already so many parent functions that they are keeping track of. I also made a long GeoGebra exploration activity that will cover most of this material without any lecturing on my part, but I'll blog about it only after I figure out how well my kids can absorb the stuff through the scaffolded (but long!) activity. Wish me luck!!

Algebraic Prism

Here is a worksheet I gave to my H. Geometry kiddies today. It was something I had made for my H. Algebra 2 kids last year, but now slightly modified to remove the function notation, specific questions about the degrees of functions, etc. The L-shaped prism looks that way entirely on purpose: Can your kids still properly identify the base shape and the height when it's an "irregular" shape, lying on its side?

I introduced to them the "box method" of multiplying polynomials, and they loved it. (You algebra teachers know what I'm talking about. Way better than drawing distributive arcs any day!)

(#4 of Pg. 2 is a little ambiguously phrased, because I want kids to remember on their own that the easiest way to find volume, once you have the base area, is to multiply by the height. They then later show that there is a different way to come up with the volume, and that it gives you the same formula in the end. See #6 on Pg. 3.)



They absolutely loved it (...It's always funny what gets kids going...), and we got into some pretty good geometric discussions about how surfaces on opposite parts of the prism correspond in area (ie. Area of DKMF = Area of CJIB + Area of AHNG), and how a little geometric intuition can save us a lot of work when calculating algebraic surface area!

Income Tax Unit (Not All Mine!)

Sam Shah had asked, so here it is -- some piecewise lessons I've done in past, leading up to a debate on tax policies at the end of the income tax unit. This course I had only taught once, two years ago, as part of the NYC Math B curriculum. (Or, loosely following the Math B curriculum, I guess.)

Disclaimer: I didn't come up with all of these materials! Some of it I did (all of the extensive scaffolding, more or less, and some word problems). My colleague and friend, Tim Jones, from my previous school had taught the honors version of the same course before me, so I took a lot of the income tax stuff from him. (I had to add a lot of scaffolding for my regular 10th-graders though, and the unit ended up taking twice as long as his class did.)

Link to piecewise unit and link to income tax unit, dumped into viewable directories.

All the lessons are numbered in order (evidence to my organizational tendencies), so they should be easy to follow, but in some cases you'll find that I took two steps forward and one step back, because I had realized that my inherited (honors) version of the lesson was just too-much-too-fast for my kids. Have fun probing! If you have questions or comments, feel free to shoot them my way even though I don't teach much of this stuff anymore.

Piecewise Functions Scaffolding Up the Wahzoo

Out of habit (aka. a compulsive need to organize my work), my lessons are always digitized and numbered sequentially/labeled with keywords, collected into chapter folders and further grouped by classes. At the end of every school year, I zip up the entire folder tree and archive it in multiple places, labeled by year. At the beginning of the school year, I download all of the past archives onto each computer I might be lesson-planning on (one at school and one at home), so that when I need to look up an old lesson, it'd be in front of me in a matter of minutes, complete with handouts and relevant quizzes and tests.

So, anyway, the other day I started to teach my Precalc kiddies the basics of piecewise functions. After the first 1.5 lessons or so (in which I had introduced the mandatory Christmas bonus breakdown in El Salvador and had asked the kids to graph the bonus and to graph a simple progressive tax system*, and also had gone over the basics of evaluating a piecewise function), I thought of digging through my old stuff to pull up some scaffolding material for writing piecewise function equations.

I pulled some of my favorite scaffolding things together into one packet this year. Here it is (I threw a couple of word problems in there for fun, since this packet I intend on grading as a project):








I am posting it because maybe you'd find it helpful to see how I break down piecewise equations into, well, pieces! (If your kids don't need these skills broken down quite so much, you might want to check out Sam Shah's collection of piecewise worksheets, which is a little more comprehensive in the skills they need. Again, I make no claims that my worksheets are comprehensive! They just step in and break it down a little more, like the problems in the textbook don't do...)

*Speaking of which, funny thing about tax systems: I randomly wondered the other day whether there are any countries in the world that actually implement a flat tax. Found this article that talks about how Estonia and two of its neighbors implement the flat tax system, and some of its less-apparent benefits and drawbacks. (For example, did you know that when neighboring countries all implement flat tax systems, it becomes a negative competition sort of thing, where each country may keep lowering their taxes, in hopes of staying competitive?) I shared some of the interesting tidbits with my kiddies, and even they thought it was fun!

And, coincidentally, El Salvador's own tax brackets are exceedingly simple. I am going to give it to the kids sometime next week, as additional practice for making graphs and writing equations.