Tag Archives: congruence

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?

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.