whY is X minus H & plus K?

We all have our methods of helping students gain an understanding of this:


Let’s start simple as most of us do in the classroom.  In years past, I (and many others) have helped students explore this concept with something like:

Function transformation Intro Activity

This concept develops as students explore more.  They learn that operations inside the primary piece of the function have horizontal effects, while outside correlates to vertical changes.  Easy enough.  But then, it happens.

Student: “Mr. Butler, I get that plus or minus moves the parabola.  But how come when you do vertical is goes the way it’s supposed to, and sideways is like opposite.”
Butler: “That’s a great question.  What do you mean by opposite.” (trying to get the student to articulate more specifically what he/she is trying to say while I ponder it myself in hoping to come up with an appropriate answer).
S: “Like vertical stuff goes up for plus, down for minus.  Just like the graph”
B: “What do you mean the graph?”
S: “The axis.  The y-axis.  But then for the other axis, the x-axis, it moves opposite.  Like negative is left, but when it’s minus it goes right.  And positive is right, but when it’s plus inside the parabola goes left.”
B: “Who else sees the pattern, or something like it?”

Lots of Hands.

B: “Well, you see whenever its inside, the transformation is opposite.  We’ll see more of that when we look at multiplying too.”

Cue the tilt head, “Huh?” OR “Oh …..I get it,” fake voice.

Credit: LIz West via Flickr
Credit: LIz West via Flickr

This process with examples didn’t answer the question conceptually, it just confirmed that there’s a pattern.  Yes, inside opposite, outside normal.  But why?

There’s an opportunity here to go deeper.  Something that I explored with students was the idea of changing inputs and outputs.  I’m pretty sure you’ve heard of the function machine:

Credit: Chris Robinson via 101qs.com

Students get this, kinda.  You choose an input, do some math, then get an output.  It’s helpful to reference the inside and outside as add-ons to the machine.


Students can kinda get a grasp on that, but it’s still abstract.  So we iterate with a few different function machines: quadratic, cubic, and maybe even some square root functions.

gifsmos gifsmos (1)

We often lean on words like shift and stretch, and then that whole confusion with the opposite comes in.  What is the opposite of stretch?  Do we say compress?  Can we compress by a negative?  What does that look like?  In this case many would favor more appropriate vocabulary like translate, reflect and dilate.  Either way through repetition and memorization we hope our students can eventually break down the effects of a,b,h and k for:

do we really need a b?

Wait, do we really understand what’s going on with the inside/opposite confusion yet?

Finally, Some Insight with the Inside

When we are making a change to the inside, we are making a change to the input, before it goes into the function machine.  That’s like making a change to all the inputs at the same time.  We conveniently have an organized visual set of the inputs: the x-axis.  So any change to the inside/input is actually a change to the x-axis.  Play with this and see if it starts to conceptualize a little better for you now.

The Applet


I’m not opposed to starting simple. In fact, I’m all for it.  There are so many layers in function transformation and it takes a while to handle what’s really going on under the hood.  I do suggest that once that question of “Why?” is primed and ready, get the students to explore with this.

Systems of Expressions

Recently I read a post by Bob Lochel (low-shell).  His stuff is great and he is one of the desmasters I’d turn to with any burning questions or wonderings.  He brings to light some simple ideas that have been playing in my head recently.  Let’s ask the bigger question:

What do these even look like?

CodeCogsEqn (10)
CodeCogsEqn (9)
CodeCogsEqn (8)

CodeCogsEqn (7)
CodeCogsEqn (6)
CodeCogsEqn (5)

CodeCogsEqn (4)
CodeCogsEqn (3)
CodeCogsEqn (2)

CodeCogsEqn (11)

I feel like I’ve often looked at all of these relationships as completely different, nearly unrelated.  The bigger picture gives us two expressions, each on the side of a comparison symbol I remember from undergrad referred to as the trichotomy.  Each of these expressions can be input into @desmos, and then the greater, equal, lesser relationship can be visualized.  I’ve put together a handful here:

Absolute Value vs Constant a|x-h|+k [?] c
Linear Vs Constant ax+b [?] c
Linear Vs Linear ax+b [?] cx+d
Quadratic Vs Constant a(x-h)2+k [?] c
Cubic Vs Constant (x-h1)(x-h2)(x-h3)+k [?] d

Some Previews:

Less, More or Equal? Where?

I don’t want to shy away from the algebraic process in articulating where these relationships are precisely defined.  I’m just a fan of see what you’re doing.  I’ve come to understand that students Believe it when they See it, and Own it when they Control it.  Supporting an answer with visual information only strengthens the understanding.  Desmos and Geogebra have make this possible.

Quadratic Formulae

Quadratic Amy
courtesy of Samuel Winter via flickr CC-BY-NC-SA License
How would you graph?   QuadraticToGraph

Would you factor?  Would you complete the square?  Quadratic Formula?  Are those my only options?

The idea of multiple representations  of the same thing, or multiple methods toward the same end, has strong correlation for learning (and teaching) mathematics.  Jo Boaler demonstrates this in her number talks.  Dan Meyer and Robert Kaplinksy discuss these through the open middle problem design. One way we see this in polynomials (including quadratics) is…


There’s the famous XBox method, often abbreviated to a formulaic method by teachers.  I like this approach, especially because it incorporates area models, grouping, flexibility for multiple values of the lead coefficient “a”.    James Key (aka @iheartgeo), brings in more of the number talk by using powers of 10 in addition to powers of x

Here’s a quick preview of his vine that relates:

XBox Numerical


Transitioning to more algebraic/abstract examples seems simpler when relating quadratics to numerical examples like this vine. He looks at the factors and the relationships between the terms.  Students in his class are prescribed a formula, but asked to make sense of the relationships and then transition to methods with more regularity.

Complete the Square

A method often confusing to Algebra 1 students because of its abstract structure is again made simple with looping short videos.

Courtesy of @NatBanting
Courtesy of @NatBanting

Plenty of teachers try to support their process with visualization, all that is intended here is to start with the visual reasoning and pattern making, then move to abstract.

The hybrid

It’s not exactly completing the square.  It’s not exactly factoring and using the roots.  It’s the compromise between the two.

  • plot the points (0,0) and (-b,0)
  • shift vertical c units.

Let’s make that easier.  Here’s a dynamic version:


Open up the Middle

But wait, the two points (x1,0) and (x2,0) that plot on the horizontal axis can be any combination that adds up to your linear coefficient “b”.  The vertical shift just needs to balance those numbers with c.

So for our example: QuadraticToGraph, the 8x needs to be split, and c needs to be balanced with the shift:

x1 x2 c shift?
8 0 20 20
0 8 20 20
4 4 20 4
2 6 20 8
1 7 20 ?
-2 10 20 ?
3 5 20 ?


Now with that bit of intuition, we can open up the middle and have the user do some exploration.  Here’s a question that encourages looking at multiple representations and seeing how the values are connected:


When thinking of factors, roots, and vertical shifts, the relationship becomes more dynamic.  We could revisit the open middle prompt and ask to define / express the relationship between a,b and c.   Struggling students can simplify the algebra using Jame’s Keys approach with substituting a value for x.  Here’s a Geogebra applet to help visualize the multiple representations:


 Play with the applet and notice that as long as the quadratics share the same line of symmetry, only a vertical shift is needed to overlap.  This also reveals the congruence in some quadratics.  Ask yourself what makes the quadratics congruent?  

Here’s a similar interactive graph built with Desmos.

What now?

I would expect that this more flexible visualization allows for more depth in understanding how a quadratic function works, both visually and algebraically.  Students who are asked to match quadratic algebraic expressions may be little less intimidated by starting with some dynamic pictures.  So now let’s return to the original question (that has multiple possibilities):

How would you graph?   QuadraticToGraph

What other questions might you ask?

Segment Similarity: Circles Part 4

So this happens in geometry:

Proportional Triangles: http://goo.gl/RDkMQo

It relates back to dilations.

Transformations Applet: http://ggbtu.be/m125888

The big idea: similar triangles have proportional measures.

Scene Change

And then we see this:

Segment Measures in Circles

And formulas like these:

Circle Segment Theorem 1
Circle Segment Theorems 2 and 3

Wait, did you see that?  Down at the bottom with the big red arrow.  It’s fine print but it looks like a proportion of some sort.

Notice and Wonder

Max at MathForum shares the awesome of the notice and wonder practice in a math classroom.  This is what I was experiencing when I saw (a)(b)=(c)(d) and some occasional references to a proportional measure of  .  I was noticing the proportions and wondering about  similar triangles, dilation and those proportional relationships.

So This was my first date with Geogebra.  I was hooked, and I never looked back.  Geogebra allowed me to visualize my intuition and confirm my conjecture.  Eventually I came up with an applet.


Flipping the Triangle: APPLET

It wasn’t enough to just show how flipping a triangle could reveal the proportional relationship with similar triangles.  We need to also build those proportions using actual measures.  I made it so that the user could drag text boxes that correlated to the needed measures from the circle.

Where’s the similarity?

But wait, are all the relationships the same then?  Not exactly.  I would guess that this is also why most texts/reference guides give 3 separate theorems.  What each instance does have in common is there are two triangles, one similar to the other.  We just need to find the reflection that reveals the proportional relationships.


I can imagine others thinking right now, “That’s cool and all, but I don’t have tech for each of my students,” or “I’m not really a techie person.”

So use paper. (Patty paper works great).  Or transparent stuff like sheet protectors.

Paper Practice

Let’s be real.  This is great for conceptual understanding and visualizing the underlying relationships, but we need to practice.  And practice often works hand in hand with paper.  So we work with this concept for a half, maybe 3/4 of a period in the applets and then we practice fluency.  I actually go back to something like the Kutasoftware sheet from the beginning.  The difference this time is that we color code the segments, redraw them as embedded similar triangles, and use proportions.

And not once did we need those similar, yet not so similar theorems for circle segments.

Clock Tangents: Circles Part 3b

Recently, there was a post looking at how to use the relationships of segments that are tangent to a circle to calculate more than just congruence.  Somewhere, and I’m not sure where, I remember Regan Galvan putting out there a tweet/post/comment about tan-tan angles on a clock.  Upon searching for it I found this:


I wanted to know how this thing worked.  So I built my own.


I purposely left out a lot of the information. I didn’t give any measures for angles or segments. This would be something that is developed in the classroom. The user and/or facilitator should be developing questions like:

  • How do I measure the angle between the clock hands?
  • How far away is the intersection point?
  • Is the intersection point always outside the circle?
  • What time is it when the two hands are perpendicular?
  • What time is it when the two hands are opposite rays?
  • Besides 12:00, what time is it when the two hands overlap?

Getting students to investigate these and asking to show evidence and create an argument to support their claim would be so much more fun than just calculating pre-made samples.  For students wanting more feedback and support, they can use the visnos interactive application.

Satellites Spinning Round: Circles Part 3 (Tangents and Incircle)


Have you ever wondered how far a satellite is from the planet?  Sometimes people think about these objects flying through the sky and rotating around a planet.  Such wonders have even inspired greats like Isaac Newton and Kepler.

Some good folk over at illustrative mathematics put this whole satellite into some simpler geometry with circles, triangles, and tangents.



Inspired by these I sought out to simplify the idea of tangents and how they fit around a circle.  We played with this:




Warm up with some questions about the applet,
    • What shapes do you see?
    • Which measures, do you think are congruent?
go into some more typical math questions,
    • What is the perimeter of the triangle?
get a little geeky
    • What is the total area of the triangle (circle included)?
Then get all geeky advanced.
  • How do you calculate the value for the area of the blue shaded area (circle excluded)?

 Student discourse:

Along the way here were some awesome teachable moments:

T: What kind of segments do you see?
S1: Tangents
T: How do you know?
S1: ’cause it says ‘tangent’.
T: I’m still not convinced. Maybe I mislabeled it. Prove it.
S2: They’re perpendicular. Tangents are perpendicular.
T: Anything else? Any other types of segments or is that it?
T: What about inside? Any segments inside the circle?
S3: Radius! There’s a radius!
T: Where?
S3: Here?
T: Where? Pretend I’m blind.
S3: You already are blind Butler, that’s why you wear glasses.
T: [fake smile]
S3: Here at AJ. Ohh, and also AF. And these ones to but you didn’t label them.
T: Do you notice anything else? What if you drag around that red thing?
S3: They’re all the same?
T: What are the same?
S3: The radiussus. Does that mean they’re equal?
T: What do you think?
S3: Yeah, equal. They have to be, cause of the red thing.

S: Hey Butler, is that a kite?
T: It sure looks like it huh? What do we know about kites?
S: There’s stuff that’s the same. Like congruent.
T: For example…?
S: The pieces that match.
T: So how can you use that?
S: They’re equal. Oh wait, they’re equal. Sooo….
And then the student goes on to apply that understanding to fill in the blanks for the missing segment lengths.

The idea of using a few measures, congruent relationships, and some triangle calculations helped the students apply the geometry and calculations further than just one calculation.  As a follow up assessment they completed a static version of the question.  This was easy to create by just taking a screenshot of the applet at some particular balance.

Some day I want to advance this further with some clock math.

Circles Part 2: Angle Relationships

Once similarity intuition has been built with circles, we can start getting into more specific relationships with angles and segments.  This post will look at using visual information from central angles and inscribed angles.

Pi Charts


Students sometimes lack intuition for the measure of something.  Andrew Stadel has developed this idea into a thorough curriculum on estimation. In my classes we started reasoning through similar exercises.  Once we had a decent understanding of circle parts and whole, we moved on to other types of angles.

Non-Central Angles

At this point most students have the common sense that a circle has 360 degrees, and a triangle is half that at 180 degrees.  Built with this intuition in mind, we look at a triangle created by inscribed angles.


The next day we get to see the formula that collapses 3 ideas down to 1.

Dynamic Angles in Circles

So many of us have tried to organize all the formulas that go with circles.  It’s nice when textbooks or other curriculum organizers dom something like:



This is nice, but it still implies that all these relationships are discrete.  Instead it would be better to look at them as connected.


I originally thought students would look at this and see how the measures relate.  It wasn’t obvious, so I interjected with a simple exercise to the students.

Congruent Overlap

I saw two half sheets (8.25×5.5) of paper on my desk.  I held them up and asked the class, “Are these congruent?”  The response was something like, “Uh, Duh Mr. Butler.”  So I challenged them further, “Would you bet $1,000 on it?”

Now the silent sound of a gambler’s contemplation…

A more valuable decision requires some precision and accuracy.  As they were thinking I allowed additional information, “You can’t touch them, but you can ask me to do anything you want.”  They took the bait.  “Just overlap them, Butler.”  Now they’re headed in the right direction.  Overlapping layers can reveal congruence or equal measure.  Anyone could ask a similar question for an image like these puzzle pieces:

Congruent Puzzle Pieces

To support the visual information, students need to look at the measures involved and start building the relationships (SMP7 and SMP8).  After multiple true versions of the math relationships, we can return to something like the original table.

Students start to see that the table is simply 3 snapshots of this dynamic relationship. Then it is easier to organize the information in their relative schemas.  From here we go to paper practice with simple examples and eventually into a more complex question with multiple parts.



I took a discrete snapshot from this applet and used it as a paper assessment.  Students were solving for complex relationships with multiple layers, and not all with the same process.  Common Core math like this requires utilizing big picture relationships while at the same time being able to articulate the specifics within those relationships.

Next time:

This series of dynamic relationships within a circle continues next looking at segments, first with only tangents, and then into various combinations with secants and/or tangents.



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