Connie Schrock and I recently presented a webinar on Great Tasks:
What is a “great task”? How is a great task used effectively in the classroom? Published by the National Council of Supervisors of Mathematics (NCSM) with support from Texas Instruments, NCSM Great Tasks for Mathematics 6-12 provides opportunities for students to explore meaningful mathematics by challenging their thinking about essential concepts and ideas. Explore tasks that are aligned to the Common Core content standards, with special attention given to the Common Core Mathematical Practices. Learn how tasks can be enhanced with the TI-84 Plus family and TI-Nspire™ technology to promote mathematical reasoning and active engagement for students of the 21st century.
I shared a story about using one of the Great Tasks with my students, which I have included in this post.
My geometry students are in a unit on Geometric Measure and Dimension. Last week, I gave them the first part of the Infinity Pizza task. Students are asked to create a fair method for cutting any triangular pizza into 3 equal-sized pieces of pizzas. I asked students to work alone for a few minutes before they started sharing what they were doing with others on their team. I walked around and watched. Many students started on paper.
I like seeing the progression of ideas from this student.
Of course at least one student had to decorate her pizza with pepperoni.
My students and I use TI-Nspire Navigator to help manage our classroom. I took a Class Capture to see what students were doing on their handhelds. Last year I read Smith & Stein’s Five Practices for Orchestrating Productive Mathematical Discussions.
The practices for teachers are to anticipate how students will respond to a task, monitor while students are working on a task to select which student work should come out in a whole class discussion and sequence that student work so that students will ultimately be able to connect the mathematics in the different solutions.
Let’s look at the second class capture I took – I can update the Class Capture as often as I want manually by pressing the Refresh button, or I can set it to update automatically every 30 seconds, which I often do. Whose work might you choose to discuss first if you were leading the class discussion?
I chose Alex first. Alex proceeded to tell us that he joined midpoints together and ended up creating a pizza with four equal slices instead of three. Alex answered a different question, but he gave us some good mathematics to remember about the triangle formed by the midsegments.
Next we moved to Kristen. Kristen was in the process of trying to trisect the angle. She hadn’t made it to whether the triangular pieces were equal in size.
Then we moved to Reagan. I made Reagan the Live Presenter, which meant that her calculator and interactions with the calculator were showing at the front of the room. Reagan told us that she created the medians of the triangle to cut her pizza. You can see from the measurements that she took that the pieces are equal in size. Reagan had hidden the lines that contained the medians to make her pizza look better, but it is by seeing the six triangles formed by the medians that she was able to explain why her three triangular pieces of pizzas have equal area.
I called on one other student to share his solution – but I have to show two other pictures that I had taken of students working on paper.
Flynt had thought about using the centroid to divide the triangle into three equal areas, and he had also noted that he could partition the base into three equal parts. One of the CCSS Standards for geometry talks about partitioning a directed line segment into a given ratio. We worked on that standard in our last unit, and it really excites me for my students to attend to precision with the language that they are using during class. I’ve been teaching geometry for 20 years, and my students have not used that language in the past.
Benny also talked about partitioning. He suggested that we needed to partition the base of the triangle into a 1:1:1 ratio.
I asked Benny to share what he had done with the rest of the class. He did, and one of the students asked whether it mattered which side of the triangle he used as the base. Benny said no, but he wanted the image rotated (which is quick and easy thanks to our interactive whiteboard) so that the second side he showed as the base would still be horizontal, and on the bottom. That’s the image you see on the bottom right.
You can see that Katlyn was using the same method as Benny with technology; however, if we look at her file, we can tell that she had not yet figured out how to trisect the base. We always have more to learn.
There is a part 2 for the Infinity Pizza task that I haven’t had time to try with my students yet. Every year, though, they come knowing more mathematics than the year before. Maybe next year we will get to Part 2!
My goal for my students is not just to provide opportunities for them to learn the content standards and use the math practices while they are working on a task – but for them to begin to recognize when they are using the practices. I gave my students a list of the practices along with a short explanation of each practice at the beginning of the year. I have that same list posted in the front of my room. This language has become part of how we talk to each other about what we are doing in class – and what our goals are for learning in each unit. Which math practice would you expect to see your students demonstrate with the Infinity Pizza task?
I asked my students which practice they think they used the most during this task. They completed their responses on a Google Form.
AK: “In this challenge, we used many of the Math Practices, but the one that sticks out to me is construct viable arguments and critique the reasoning of others. I think this was most prominent because of the way each student was given the opportunity to share their solution and the class in turn was given the opportunity to compare and support that student’s method or argue for a different approach. Either way it provoked in-depth thoughts to look beyond the simple “find the solution and move on to the next problem”.”
Katie noted, “I chose [reason abstractly and quantitatively] because when you have a problem like this it’s easier to look at numbers and general terms. I first had to think about the question. … It was hard trying to think of where to start, so I thought about cutting the pizza with the lines connecting the midpoints together. This would have worked had the quadrilateral these lines made been a square, but it wasn’t because I was using a right triangle. So I went to look at the lines of medians. Well, the area for each triangle then was not equal. This was trying my patience, but I kept going. I then looked at the circumcenter and the lines connecting that, but it ended up just like the first one, I had three triangles and a quadrilateral. Which was four pieces that definitely did not have the same area. Then I just thought for a few minutes. It then dawned on me that we were dealing with area and the formula for the area of a triangle is 1/2bh(where b=base and h=height). So, I thought how can I get three triangles from one where the base and height never change. I then used 21 for my base of the triangle because 3 will go into 21, 7 times. I had three triangles with a base of 7 and the height couldn’t possibly different because they all came from the same triangle.”
Chandler: “I felt that I mostly modeled with mathematics throughout this activity because the entire goal of Infinity Pizza was to utilize a certain math practice (or more than one) in order to determine how to evenly split a pizza into three perfectly even pieces. … Since I knew that the centroid would give me the point of central mass, I searched for the centroid by drawing a line from the perfect center of each angle to the exact median on the opposite side of the triangle. … Note that this was conducted on an equilateral triangle. Next, to elaborate, I tried the same procedure on an isosceles triangle and a scalene triangle. The results were conclusive — my method was feasible for any triangle despite any side length differences, even if the pieces’ sides were of different lengths. …”
Kelsey: In this activity you had to construct viable arguments and critique the reasoning of others the most. The reason I say this is because you had to come up with a way to split the pizza into 3 equal slices and you had to persuade the class that your reason worked everytime. Also you had to listen to other peoples reasons why and critique if their answer was wrong or write.
Roselynn chose make sense of problems and persevere in solving them: “In order for us to solve this problem, we had to have perseverance. This wasn’t a problem where the answer took a minute or two to solve. It took us a good portion of the block to construct our argument. It also took many approaches. We didn’t get a good answer the first try. It took a while to make sense of the problem, and we had to persevere in solving it.”
And so the journey continues, trying to provide students great tasks that help them make sense of mathematics …