On April 25th, 2019, I visited the Environmental Sciences Magnet School at Mary Hooker (ESM) to guest-teach a sixth grade math lesson on volume in real world situations. Prior to my visit, the students had been practicing solving the volume of rectangular prisms using cubic units with fractional edge lengths.
The learning objective of my lesson was as follows:
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V= lwh and V= bh to find volumes of right rectangular prisms with fractional edge lengths in the context in the context of solving real-world and mathematical problems.
These objectives are outlined in the Eureka Math Instructional Guide, which is the math curriculum used by ESM.
My lesson plan which I carried out was the following:
- 3 Act Task – Act 1
I opened the class with only a brief introduction, explaining to the students that we would be watching a video two times. I instructed them to write down things they notice and things they wonder about the video in the boxes designated on their worksheet.
2. Notices and Wonders
I proceeded to write two columns on the board labelled “Notice” and “Wonder” I asked the students to share some of the things they had written down. I had the goal of the students “wondering” the main question for the day which was “How many Rubik’s cubes are needed to fill the box?” and to my delight, it was the first of the wonders that the students shared
3. Main Question and Estimates
Next, I asked the students which of our wonders they thought would be the best main question to investigate. Again, they were right on their first try, probably because they knew this was a volume lesson, and selected “How many Rubik’s cubes are needed to fill the box?”
Once this had been decided, I asked the students to make an estimate of what would be too high, too low, and about right for how many Rubik’s cubes would fit. The students took a lot of time on these, examining the actual box and Rubik’s cubes from the video.
4. 3 Act Task- Act 2
The next question I asked the class “What information do we need to answer our main question?”. I made a list on the board again as they called out answers, which were all in the realm of saying that the dimensions of the box and Rubik’s cube were needed. I then put up the slide displaying these measurements, and Mr. Smith suggested to the class that they sketch the box and the cube and label the side measurements. From there, I told the class that they had all the information they needed to answer the main question, and I asked them to start trying out solving.
5. Student’s Explanations of their work
Once students had had a couple minutes to work on solving without help, I began to move around to check-in with the students and draw their attention to potential mistakes. In my first two experiences with this particular class, it seemed like most kids had a fairly thorough understanding of the lessons’ content and were very practiced at the skills in the objective. This time, the kids were definitely in different places in terms of their understanding of volume, so I had some students finish in under 10 minutes, who were instructed by Mr. Smith to do it again without the calculator. As further assessment beyond just me checking in with students, I asked students who completed the problem to head to my camera in the back of the room, and hold up their sheet and explain how they know how many cubes will fill the box.
6. 3 Act Task- Act 3
Admittedly, we did not formally reach Act 3, which would have involved revealing the answer and reviewing the estimates. Unfortunately, due to students being in different places as I previously mentioned, some of the students were working on the 3 Act task until the very end, and I did not want to give away the answer. Instead, once students had completed their calculations and explained their work to my camera, I quietly let them know they had successfully answered the main question. Below is some of the students completed work.
7. Extension Activity
Only a handful of students reached the extension activity, and those who did probably only worked on it for approximately 5 minutes towards the end of the lesson. It essentially extended the 3 Act Task situation to ask “How many boxes of Rubik’s cubes will fill a trailer truck? How many Rubik’s cubes would be in the trailer?
My lesson was equitable in that the items used in the problem being addressed were objects familiar to the students universally, as the 3 Act Task video displayed them in the same way to all, and furthermore, the actual objects were equally available for all students to look at during class time. Thus, none of the items involved in the situation made the problem inequitable for students due to a lack of familiarity. Furthermore, I made a point to never call on the same student twice during the beginning portion of class, calling on nearly every student and writing down every response, giving value to each one of them. Finally, when I was going around the room helping students, I often found myself answering the questions to many of the same students. I noticed however that there were a handful of students who were not asking me questions, but seemed to be struggling, so I tried to help them just as much as the more outgoing students by asking them questions to try to guide their work in a direction they might understand better. Overall, I feel my lesson did a good job of being related to a familiar topic to all students, and that I did well in conducting the lesson in a way inclusive and helpful to all kinds of student.
My lesson had formative, individualized, captured, assessment which was directly connected to the learning goal. It was formative in that much of the assessment took place throughout the lesson through me asking student questions about their work, and seeing and drawing their attention to mistakes made in the process. This was individualized in that I had one-on-one conversations with students, and additionally when students would go explain their work to the camera, they went mostly alone, or in pairs under the instruction that they both were expected to talk the same amount. My assessment was captured in that I have both video evidence as well as student work on their worksheets to serve as evidence of learning. Assessment was directly connected to the learning goal of using cubic units to find the volume of rectangular prism as this was exactly the solving needed to answer the main question.
While this lesson was a success, I do feel that it was less creative due to the students seemingly finding it less fun than my lesson before. However, a three act task, compared with the activity I conducted last time is an entirely different type of teaching, and it is important to have a repertoire of teaching techniques which I’m comfortable to move into, so at the end of the day, it was good to try this out in order to expand my repertoire.
There were a couple of mistakes in the flow of the lesson. While I did improve on my main self-critique in the past, which has been speaking too fast, there were other pieces of my presentation that should have been more polished. The first, thing that could have gone better was the estimates. We ended up spending 10 minutes on students guessing too high, too low, and about right estimates for how many Rubik’s cubes would fill the box, because many students were too meticulous. The estimate really should have been a one minute task where the students took a quick guess. We ended up not reaching the point of revisiting the estimates, which would have given the lesson both closure and fun as it would have allowed students to see how good their estimates actually were, which is quite exciting for them. But, in part due to taking too long making estimates in the first place, there was never time for this. Another mistake I made was in the initial presentation itself. With the technology at ESM, I had to email my powerpoint to Mr. Smith before I arrived. However, I had actually changed the side length labels slightly on the box (the students would never actually measure the box itself so approximating measures was ok), so that the numbers were easier to work with. However on one of the slides in my presentation, the old measurements were labelled, causing much confusion obviously. I should have easily prevented this mistake by checking my slides more stringently before the lesson.
As a whole, many students found this lesson difficult. I think there were two reason for this. The first is that I sense some of the students still didn’t feel practiced enough with this skill, and were not confident enough in the concept of volume to be able to formulate a method to use it in a real life situation. The second is that many students were likely not used to the format of a three act task in which they are only solving one problem. I think that the typical structure of their lessons is likely more along the lines of solving a multitude of problems, and following the same steps somewhat blindly in order to find a correct answer. Finding correct answers in this typical view also likely serves are reinforcement to students that they are doing something right, helping them build confidence that they can solve this type of problem. However, this is not to say that this format of a lesson is better than a three act task. Particularly because the objective involved solving real world problems, I think the format of a three act task is excellent because, just like in the real world, students have to discover the question themselves, and not be told how exactly to get to an answer, and thus discover the process themselves as well. These students were likely less comfortable with this setup because they are so used to one where the steps they must take are handed to them. I think that with more practice with more diverse types of lessons like three act tasks, these students might become more comfortable with discovering the components of a problem, and this would make them better real-world mathematicians.