Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
What do you do when the standard for the day gives away what you want students do explore and figure out on their own?
I’ve made a deal with my administrator to post the process standard for the day (Math Practice) instead of the content standard.
In many of our geometry classes, our learning goals include look for and make use of structure and look for and express regularity in repeated reasoning.
We defined midsegment:
A midsegment of a triangle is a segment whose endpoints are the midpoints of two sides of the triangle.
The midsegment of a trapezoid is a segment whose endpoints are the midpoints of the non-parallel sides.
Then we constructed the midsegment of a trapezoid. Students observed the trapezoid as I changed the trapezoid.
I sent a Quick Poll: What do you think is true about a midsegment of the trapezoid?
It creates both a triangle and a trapezoid. 1
all the midpoints form a similar triangle 1
mn is parallel to yx 2
((1)/(2))the size of the origin 1
parallel to base of triangle 1
all the midsegments make a similar triangle upside down 1
parallel to the base 1
and a trap 1
creates triangle and trapezoid 1
It will form the side of a triangle that is similar to the original. 1
The midsegment is parallel to the side not involved in making the midsegment. 1
it would be a median 1
MN is parallel to YX 1
parallel to base 1
mn parallel to yx 1
MN is parallel to the bottom line 1
cuts the tri into a trap and tri 1
all the segments will make a similar triangle tothe original 1
all of the midpoints connected make a similar triangle to the original one 1
creates ∆ on top + trap. on bottom 1
Triangle XMN is similar to triangle XYZ.
Line MN is parallel to line YX. 1
2/3 the largest side 1
2/3 o 1
side parallel to the midsegment is a 1
it makes a triangle and a tra 1
mn is ll to yx, mnx is congruent to triangle mon 1
In order to change things up a bit, I quickly printed the students conjectures, cut them up, and distributed a few to each team. Now you decide whether the conjectures you’ve been given are true. And if so, why?
I used Class Capture to monitor while the students talked and worked (and played/explored beyond the given conjectures).
Then I asked what they figured out through a Quick Poll:
seg mn parallel to both seg ab and seg dc 1
parallel to both of the bases 1
it cre8 2 trap 1
mn is parallel to dc and ab 2
Trapezoid ABCD is similar to ABNM.
Lines AB and CD are parallel to the midsegment. 1
MN is parrellel to AB and DC 1
It creates a line that is parallel to the bases and forms two trapezoids. 1
nidsegment is parallel to top and bottom sides 1
it would be parallel to the sides above and below it 1
MN is parallel to DC and AB 1
its parallel to DC 1
It makes two trapezoids 1
it is // 2 ab and dc 1
it forms two trapozoids 1
It is parallel to the sides above and below it. 1
makes 2 trap. 1
It makes a similar trapezoid. 1
they make 2 trapezoid, ab+dc/2 1
parellel to base of trapezoid 1
ab ll to mn ll to dc. when a parallelogram, creates 2 congruemt trapezoids 1
trapezoid ABNM is similar to trapezoid MNCD 1
all midsegments make a diala 1
And then we talked about how they knew these statements were true.
Jameria had a lot of measurements on her trapezoid. I made her the Live Presenter. What conjectures can we consider using this information?
I made Jared the Live Presenter.
What does Jared’s auxiliary line buy us mathematically?
I made Landon the Live Presenter.
What conjectures can we consider using this information?
I sent a Quick Poll to formatively assess whether students could use the conjecture we made about the length of the midsegment compared to the length of the bases of the trapezoid.
What about the midsegments of a triangle?
We had not yet started our unit on dilations, and so there was more to the why in a later lesson.
And so the journey continues, even making deals with my administrators as needed, to create a classroom where students get to make and test and prove their own conjectures instead of being given theorems from our textbook to prove.