One of the most important new concepts in algebra 2 and beyond is the notion of *transformation*. That is, given the function *f*(*x*), we can change any function’s position and growth by using the same instructions, much like giving directions from a map.

I’ve just introduced functions at this point in the calendar, so I’ve designed this activity to reinforce f(x) as a rule, that once a mapping is created, the mapping holds for all subsequent calls.

So just create a random table, one that’s simpler than anything I’d do in class. (One of the incredibly irritating things about blogging is that it’s insanely time-consuming to create images for publication that take next to no time at all to do on a smartboard, but I never think of capturing images while on a smartboard.)

x | f(x) |

-3 | 2 |

-1 | 5 |

1 | 6 |

3 | 3 |

5 | -1 |

That looks like this:

So then I ask if this is f(x), what would f(x+2) look like? Someone brave will always say “Two to the right”.

At that point, I always say “This is a totally logical guess and one of the most annoying things in math from this point on is that your guess is wrong.” (I originally developed the concept of a parabola as the product of two lines as another way of explaining this confusing relationship. Confusing to normal people. Mathies think it makes sense, but they’re weird.)

I add a column to the table. “We start with x. Then we add 2. **Then** we make the function call. Note the function call comes after the addition of the value. This is important. Now, we have three columns, but we are starting with our x and that’s still our input value. We graph it against the outer column, the output value for f(x+2).”

x | x+2 | f(x+2) |

-3 | -1 | 5 |

-1 | 1 | 6 |

1 | 3 | 3 |

3 | 5 | -1 |

I’ll ask how we can bring the -3 back in, and after some mulling, they’ll suggest that I add -5 to the table. So I add:

-5 | -3 | 2 |

to the bottom. But I’ve been plotting points all along, so the kids can see it’s not going as expected.

“Yes, indeed. I’ll be teaching this concept in many ways over the next few months, and I ask you to start wrapping your head around this now. We have many ways of envisioning this. When working with points as opposed to an entire function, it might be helpful to think of it this way: Suppose I’m standing at -3, and I want to add two. This has the effect of me *reaching to the right* on the number line and pulling the output value back to me–to the *left*, as it were.”

I go through this several times. Whether or not students remember everything I teach, I always want them to remember that at the time, they understood the concept.

“So if standing on -3 and reaching ahead is addition and move the whole function to the left, how would I move the whole function to the right?”

If I don’t get a ready chorus of “subtract?” I know that I need to try one more addition example, but I usually get a good response.

“Exactly. So let’s try that.”

x | x-2 | f(x-2) |

-3 | -5 | NS |

-1 | -3 | 2 |

1 | -1 | 5 |

3 | 1 | 6 |

5 | 3 | 3 |

7 | 5 | -1 |

One year, I had a doubter who noticed that I’d made up these numbers. How did we know it’d work on any numbers? I told him I’d show him more later, but for now, imagine if I had a table like this:

x | f(x) |

1 | 1 |

2 | 2 |

3 | 3 |

etc.

Then I told him, “Now, imagine I put decimal values in there, fractions, whatever. Imagine that no matter how I change the x, the new value has an entry in the table and thus an output. So imagine I added 50. There’d be a value 50 ahead that I could reach forward or backwards.”

“In fact, we’ll eventually do all this with equations that are functions, instead of randomly generated points. But I start with points so you won’t forget that it works with any series of values that I can commit math on. Which isn’t all functions, of course, but that’s another story.”

“But if adding makes it go left and right, how do we make a function go up and down? Discuss that among yourselves for a minute or so.”

Sometimes a student will see that we’ve been changing x so far. Otherwise I’ll point it out.

“The function call itself is key to understanding this. If you change the value before you make the function call, then you are changing the input to the function. Simpler: you’re changing x before you call the function. But once the value comes out of the function, that is, once it’s no longer the input, it’s the….” I always wait for the class to chime in again–are they paying attention?

“Output!”

“Right. But output is no longer x. Output is”

“f of x!”

At this point, I call on a mid-level student. “So, Sanjana, up to now, we’ve been changing x before making the call to the function. See how the new column is in the middle? What could I do differently?”

And I wait until someone suggests making the column on the right, after the f(x).

x | f(x) | f(x)- 3 |

-3 | 2 | -1 |

-1 | 5 | 2 |

1 | 6 | 3 |

3 | 3 | 0 |

5 | -1 | -3 |

I’m giving a skeletal version of this. Often the kids have whiteboards and are calculating all this along with me. I’ll give some quick learning checks in terms of moving to the right and left, up and down.

The primary learning objective for is to grasp the meaning of horizontal and vertical translations–soon to be known as *h* and *k*. But as an introduction, I define them in terms of function notation.

We usually end this activity by combining vertical and horizontal shifts.

What would f(x-2)+ 3 look like? Well, you’d need another column.

x | x-2 | f(x-2) | f(x-2)+1 |

-3 | -5 | NS | |

-1 | -3 | 2 | 5 |

1 | -1 | 5 | 8 |

3 | 1 | 6 | 9 |

5 | 3 | 3 | 7 |

7 | 5 | -1 | 2 |

I connect them this time just to show that one point is in both the original and the transformation.

Ultimately, this goes to transforming functions, not points. That’s the next unit, transforming parent functions. I have a colleague who teaches transformations entirely by points. I start down that path (not from his example, just because that’s how this works), but the purpose of transformations, pedagogically speaking, is for students to understand that entire equations can be changed at the unit level, without replotting points. At the same time, I want the students to know that the process begins at the point level.

Over time, the students start to understand what I often call inside and outside, or before and after. Changes to the input value affect the x, or the horizontal because they occur before the function is called. Changes to the output value affect the y, or the vertical, because they occur after the function is called. Introducing this on a point by point basis creates a memory for that.

At best, this lesson functions as more than just a graphing exercise, something to introduce vertical and horizontal shift. It should ideally give students an understanding of the algebra behind it. Later on, when they are asked to solve equations like:

Find f(a) = 32 for f(x)=3(x-2)^{2}+5

Weaker students have trouble with understanding order of operations, and a memory of “inside” and “outside” the function can be helpful.

If I were writing algebra 1 curriculum, I’d throw out quadratics, introduce a few parent functions, and teach them function notation and simple transformations. It’s a complicated topic that they’ll see all the way through precalc, at least.

I’ll discuss stretch and its complexities in another post.

August 1st, 2017 at 12:59 am

[…] Source: Education Realist […]

August 2nd, 2017 at 4:55 am

Transformations only from points? I can visualize student heads exploding.

Anyway, I like how you did this. I worked on similar lines this past year, and I think I might have neglected to redo points enough for some students who in theory had done that part of Algebra 2 before.

I do some transformation in Geometry, but who is going to remember that after the summer ends? Of course it wasn’t function notation, but I think it did help somewhat, probably moreso in the stretches and reflections for those who could recall.

August 2nd, 2017 at 6:43 pm

Yeah, transformations in geometry seem totally disconnected from functions, which is weird. They shouldn’t be.

I want to reiterate that I’m really a big believer in getting kids to transform at the function level, but I want to spend some time making sure they understand that if they did it at the point level, it’d work the same.

September 24th, 2017 at 2:45 am

[…] part of our Teacher Federalism agreement, I now include the reciprocal function as one of my parent functions in Algebra 2. But time constraints don’t allow me to really dig into the […]

November 23rd, 2017 at 6:35 pm

[…] couple months ago, I used laptops and Desmos to teach transformations, and after two blocks that went….well, I suppose, I used whiteboards to do the same lesson in […]

August 14th, 2018 at 7:38 am

Hi! Came across your blog a few days ago, and while reading this post and “The Product of Two Lines” I had a realization that you might find useful (if nothing else, for keywords to google).

The backwards-ness of horizontal translations is the behavior that mathematicians call “contravariance vs. covariance”. Inside transformations are contravariant, outside are covariant.

https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

Graphically, think of it like this: the transformation x->(x+2) shifts *the coordinate axes* 2 units to the right, while leaving the function fixed… which is equivalent to shifting the function 2 units to the left.

This also shows up in linear algebra, in the definition of the change of basis matrix: https://www.math.hmc.edu/calculus/tutorials/changebasis/ it’s the reason why the effects of P and P-inverse are swapped relative to what you’d expect.

Turns out transforming the axes is a pretty fundamental idea in physics, and I encountered it a few times before realizing there was a spooky math thing going on behind the scenes. It’s why we care about upstairs vs. downstairs indices on tensors in general relativity. The statement that the translation and rotation operators are unitary in quantum mechanics is equivalent to the statement “translating/rotating a point one way is indistinguishable from translating/rotating the coordinate axes the opposite way”.

Anyway, let me know if this was interesting, or if you have any more examples of this behavior! I’m enjoying all your posts on math pedagogy.

August 14th, 2018 at 12:45 pm

That is really interesting. Thank you so much–I will read up on it and see if I can turn that into an explanation that helps my kids!