Earlier this year in Advanced Geometry, my kids were introduced to Blermions (original post from when I created the lesson; new post after I tweaked and taught the lesson). That lesson gets kids to understand a bit about cyclic quadrilaterals and some of their properties.
Now we are at the end of the year, and one of my Advanced Geometry sections had three classes that the other section didn’t have. So I had to come up with something supplemental. Thank goodness for twitter. You see, @jacehan was using my blermion activity, and some of his kids asked him “what if the circle was inside the quadrilateral?”
Of course, genius that @jacehan is [his blog is here], he named these creatures merblions.
With this one question, I had the makings of an amazing three days ready for me. You see, in Geoemtry, we had just finished studying angle bisectors (and how they related to pouring salt on polygons). We had also just finished studying triangle congruency. (I know that is usually taught earlier in the year, but when I rearranged the course, it fit best near the end of the year.) So those were two powerful tools to analyze merblions.
So I told students to pair up. And they were given the above picture and told that: “A merblion has an inscribed circle which is tangent to all four sides of the quadrilateral.”
That is all.
Then I told students that in some ways, this is a culmination of everything they’ve done all year. They have everything they’ve learned at their disposal. Geogebra. Paper. Rulers. Compasses. Protractors. But mostly, they need to make conjectures and see if they are true — either getting a lot of inductive evidence or by using deductive logic. Anything they wanted to figure out about merblions were fair game.
I also highlighted that the other geometry teacher and I started investigating these, realized they were very rich and there was a lot to discover, but we purposefully stopped investigating them. We wanted our students to make the discoveries, without us accidentally guiding them.
We also told them that they needed to persevere, and be okay trying lots of things. But if they ever felt their wheels were turning and still nothing was happening, they could call us over for a nudge. (I created a list of things I could say to kids to help nudge them along if they got stuck… I didn’t have to use it more than once! Kids were into it.) They knew at the third day, they would be presenting (informally) their findings to the class. So they had to keep track of things, take screenshots, etc.
While they worked (with music!), I saw kids make conjectures, find they weren’t true, and then move on. I then realized kids weren’t recording their “failed” conjectures. But that data is important! So I told kids to keep track of all of their ideas, and even if their idea didn’t turn out to be true, it is totally worthy of putting into their presentation! It helps us see their avenues of inquiry. Similarly, I told students to record their conjecture, even if they couldn’t prove them deductively.
The kids were doing so many interesting things — including things I hadn’t thought of. (Two pairs tried finding the smallest merblion, by area, that could fit around a circle of a given size! Three pairs tried to do an “always/sometimes/never” with “A _____ is A/S/N a merblion” where the blank were all the quadrilaterals we’ve studied [kites, rhombuses, trapezoids, etc.]. One pair noted that to use Geogebra to draw a merblion, you only need a circle and two points, but the two points couldn’t be any two points — so they wondered where those two points could be located.) It was great.
They continued on the next day, and spend the last 20 minutes of the second class throwing some slides up in our google presentation [posted here, with identifying information of students removed].
What they ended up discovering was awesome.
some big results (some proved, some unproved) found
1. The center of the circle inscribed in the merblion is the intersection of the four angle bisectors. And if we cut a merblion out of cardstock and did the “salt pouring activity,” we would see the salt form a pyramid with a merblion base and a single peak (where the peak would exist at the center of the inscribed circle).
2. Kites, squares, and rhombuses are all merblions. However rectangles are only merblions if they are squares, and parallelograms are merblions only if they are rhombuses. Some trapezoids are merblions and some aren’t.
3. No concave quadrilateral can be a merblion.
4. A merblion has two pairs of opposite angles which are acute, and two pairs of opposite angles which are obtuse (unless you have a square).
5. A merblion is secretly composed of four kites. And the four kites have two opposite right angles. (Which means that the non-right angles are supplementary in these kites.)
6. In a merblion, the sum of the lengths of opposite sides are equal.
7. The area of a merblion can be computed by finding the perimeter, halving it, and multiplying it by the radius of the inscribed circle.
8. For all merblions that can be drawn around a given circle, the merblion with the least area is a square.
9. In the other class (not in my class) students found this result… The two angles here are always supplementary.
Why I loved this
The kids were totally engaged. They didn’t feel pressure to produce “the right answer” because there was no right answer. (And no grade associated with this work.) I emphasized that all conjectures (even if they don’t work out) were valid, so kids felt okay writing anything and anything down. I didn’t have a specific outcome they had to come up with, so I wasn’t leading. Kids could do anything! They got to work together.
And when some results were presented that explained things that people were wondering about — there were noticeable ooohing and aaahing (for example, result #6!).
And after the presentation happened, it became clear to everyone that by crowdsourcing this problem, we were able to see lots of results and then start examining how the different results related to each other (so for example, result #6 explains #2).
This was very fun. Very very fun.