I don’t have the exact quote, but I read in Embedded Formative Assessment (Dylan Wiliam) that teachers having time to plan the questions that they will ask in a lesson to push and probe student thinking is important. We should ask questions to push students’ thinking forward and to uncover student thinking (and misconceptions).
I have read more about this idea of selecting questions ahead of time in Transformative Assessment (James Popham) and Transformative Assessment in Action (James Popham). Popham suggests that teachers not only need to plan what questions they will ask for formative assessment but also how they will respond when all students answer the question correctly versus the majority of students versus half of the students versus few or none of the students. I think that is ideal. I haven’t decided yet that it is practical for every lesson I teach.
In our dilations unit, we did a lesson on Pythagorean Triples and Pythagorean Relationships. We used part of the linked Math Nspired activities by the same name. The main purpose was to provide students an opportunity to make connection between a primitive Pythagorean Triple and the resulting triangles that can be dilated from that triangle. But at the recommendation of my upperclassman who have already taken the ACT and SAT, we also spent a bit of time providing students an opportunity to determine whether a triangle was acute, right, or obtuse given its 3 side lengths. While I think this concept could be implied from CCSS 7.G.A.2, it will be 2-3 more years before we have high school students who have been through CCSS Grade 7.
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
I started the lesson by asking students what question they might ask if they knew the side lengths of a triangle were 8, 16, and 17. Many asked about classifying the triangle, but a few asked what the area of the triangle would be. The question provided a nice problem solving points opportunity for those who wanted to learn about calculating the area of a triangle given the three side lengths.
I exported the questions students asked and have included below:
Is it a right triangle? 6
What is the area of a triangle with side lengths of 8, 16, and 17? 1
Does it have a right angle? 2
What type of triangle do these 3 sides make? 2
is it a right triangle 1
The question would be what type of triangle is this 1
is it scalene? 1
what is the question that goes with this information? 1
what kind of triangle is it 1
classify the triangle? 1
what type of triangle is it 2
Is it a pythagorean triple? 2
what kind of triangle is it? 1
is it a pythagorean triple? 1
What type of triangle is it? 3
is it an acute triangle? 1
with these given measurements is there a pythagrian triple? 1
which side is the base? 1
what type of triangle does this three sides make 1
Is it obtuse? 1
is it a right triangle? 8
is this a right triangle? 2
Is the set of numbers a Pythagorean Triple? 1
can we find the area 1
what is the height of the triangle? 1
is it a pythagorean triple 1
What is the height? 1
Which are the legs, and which is the hypotenuse? 1
what is the area? 1
Can the area be found? 1
how do you know what the base is 1
is this a right triangle 1
whats is the base 1
what is the area 1
is it a right triangle ? 1
what type of triangle is it? 2
what is the height? to find area. 1
As students interact with the Pythagorean Relationships TNS document, they record whether the given side lengths form a triangle that is acute, right, obtuse, or nonexistent. Then students look for regularity in repeated reasoning to describe a relationship that is true about the squares of the sides of the triangles.
For all of the given triangles, a≤b≤c. Some of my students wrote that a triangle is acute when a2+b2>c2. Others wrote that the triangle is acute when c2< a2+b2. Students already knew that a triangle is right when a2+b2=c2 or c2= a2+b2. Students also determined that a triangle is obtuse when a2+b2<c2 or c2>a2+b2. Students already knew that for 3 lengths to form a triangle, a+b>c, a+c>b, and b+c>a.
I sent a Quick Poll to assess student understanding. Our geometry team has a set of pre-prepared Quick Polls for each lesson. Teachers send them as needed in their classrooms. We don’t use every one of them, and we don’t necessarily send the same polls each time we teach a lesson. We practice formative assessment during our lessons to decide which questions to send and use the results to adjust the lesson.
I didn’t want to send three side lengths that formed a right triangle. Nor did I want to send three lengths that did not form a triangle. The first one I came upon happened to be an obtuse triangle. I sent the poll.
And I was surprised by the results. The students had determined that for a triangle to be obtuse, a2+b2<c2. Why did one-third of them miss the question? I had to think fast. I could have shown them the correct answer. And then I could have worked the problem correctly. Or I could have asked what misconception the students who marked acute had. Would everyone have paid attention to that?
What I did instead was to show the students the results without displaying the correct answer.
I asked students to find another student in the room and construct a viable argument and critique the reasoning of others. I walked around and listened to their arguments. And I sent the poll again.
At that point, the students shared what happened to those students who had marked acute the first time. They had only observed that 8+15>18 was true instead of also noting that 82+152>182 was not true. And so it struck me at that moment that I had gotten lucky. Without realizing it ahead of time, I had chosen the right problem to send students to uncover their misconceptions. Had I sent another of the prepared Quick Polls instead that asked students to classify a triangle with side lengths 16, 48, and 50, all of the students would have gotten it correct, but they would have gotten it correct for the wrong reason. For that triangle, both 16+48>50 and 162+482>502 are true, and so the students would have chosen acute even if they had incorrectly used the Triangle Inequality Theorem to decide that.
I am glad that I sometimes get lucky as the journey continues …