This task is connected to an attempt to set a world record for the farthest distance traveled rolling across swiss (exercise balls). In order to break the record, Tyler, one of the Dude Perfect members, has to use forward motion only to move his body across exercise balls that are lined up in a row. If he falls off the balls or his hands or feet touch the ground the attempt is not successful. The surfing across the balls is done with Tyler’s body being horizontal lying on the balls so it is similar to lying down on a surfboard. An actual surfboard is not used to move across the balls. My learning goal for this lesson was for students to be able to solve a real-life problem using numerical and algebraic expressions and equations. This activity was used as introduction for this goal. I first had students watch the Youtube video entitled “World Record Exercise Ball Surfing” up until the 1:39 mark of the video (Dude Perfect, 2018). This helps to motivate students in solving the task and also lets them become familiar with the task context. The video is stopped before Tyler’s attempt to break the World Record is shown. If students want more detail on how Tyler will move across the balls a couple seconds of the video starting at the 2 minute and 1 second mark could be shown. I then posed the question for the problem. How many swiss (exercise) balls does Tyler need in order to break the Guinness World Record of 255 feet and 5 inches? Students next provided a number for three prompts. -Write down a guess that you think is correct. -Write down a number that you know is too low. -Write down a number that you know is too high. The guesses ranged from 20 to 128 balls, with most guesses around 50. The majority of students felt that 20 balls or below was too low and that more than 100 balls was too high. Students then wrote down what information they would need to know in order to solve this problem. The most common information that students wanted to know was the distance of the space between the balls and the size or the diameter of the balls. I provided students with the diameter of the balls, 26 inches, and also a picture from the video. The picture shows that Tyler has already started the attempt. The first ball initially started at the strip of black tape. Students worked in groups of three to four. Students individually thought about how they would solve the problem, then discussed in their group a strategy they felt would be best and developed a solution. Before implementing the task with students I thought about possible student strategies as well as questions that I would ask students. For each of the possible student strategies, I thought of assessing and anticipating questions that could be asked for each strategy. Table 1 provides more details on these general types of questions. Assessing questions enable teachers to elicit student thinking and advancing questions position teachers to use student thinking to move students toward the goals of the lesson. Further, the purpose of advancing questions is to support students in moving forward in solving the task beyond their current thinking or to explore underlying mathematics more deeply. There are different solution strategies that I anticipated that students would use. This was possible because the question did not state or imply any solution strategy. In order to answer the question students must justify why their approach makes sense. For each strategy, I have listed assessing and advancing questions that are based on student thinking and how to move this forward. If students just guessed a number of balls the questions focus students on what information then can use to solve the problem mathematically as well as looking at cases with a small number of balls to develop ideas for how to work towards the number of balls needed for 255 feet and 5 inches. In the numerical expression strategy students are not considering that there is one less gap than the number of balls being used. The first advancing questions help students to see this. Students are also asked if they can come up with an equation and what would happen if the World Record distance increased. The questions for the algebraic equation ensure that students know what each part of the equation represents in the real world context as well as considering how different lengths would affect the solution. In the actual video the gaps between the balls are not all exactly the same so the first advancing question has students think about why the actual number of balls might not match the answer. In this strategy students also estimated the length of the gap which could affect the accuracy of the answer. The questions for the proportional reasoning strategy are similar to the algebraic equation questions with some different assessing questions to determine how the length of the gap was calculated.
When I implemented this activity, there were a few errors that students made that I had not planned questions for that could be added to Table 2. Some students used a numerical expression strategy but were working in different units, feet for the World Record distance and inches for the gap and diameter of the ball. The groups that did this were able to identify their error when finding the number of balls they calculated was not reasonable. If a group did not notice this they could be asked if working in different units affects their answer. One group decided to work in feet, and wrote 255.5 feet as the current World Record distance. I asked this group what half a foot was in inches and they were able to notice the error. Most of the groups used a numerical expression strategy and tended to use an estimation of 36 inches for the length of the gap between the balls. These groups felt that the gap looked like the size of a yard stick. For the groups that used a numerical expression strategy, I used my advancing questions so they would consider that the number of gaps should be one less than the number of balls. Students were able to understand this part, but had difficulty in translating this to how they would adjust their numerical expression. The groups were unsure if they should add or subtract the length of the gap to the current World Record distance. The rationale to subtract the length was due to the fact that there was an extra gap that needed to be removed from consideration. To do it correctly, the extra gap gets added because the last ball is at the World Record length and the last gap would then go 36 inches beyond this length. In order to look at this more I decided to build on the numerical expression strategy used by many groups to discuss how this strategy could be transitioned to be shown with an algebraic equation. There were not any groups that developed an algebraic equation. I used 36 inches for the gap since many groups had done this. Students agreed that we were trying to find the number of balls so we defined x as the number of balls. One group’s strategy that I had not anticipated was to take the World Record distance and divide this by 26 inches to find the number of balls without any gaps so I started with this equation, 26x = 3065. I then asked how we could incorporate the estimated 36 inch gap knowing that there is one less gap than the number of balls. I followed up with asking if there are 50 balls, how many gaps would there be? If there were 42 balls how many gaps? If there were 40 balls, how many gaps would there be? If there were x balls how many gaps? Students were able to generalize that there would be x-1 gaps. I then connected that students had added the diameter of the ball to the gap estimate in their strategy so the equation would be 26x + 36 (x -1) = 3065. In solving the equation, it was seen that the length of the gap is added to the current World Record length. Next, we watched the rest of the video to see that Tyler ended up breaking the World Record. To tie the World Record there were 41 balls used so 42 balls broke the record. In total Tyler used 47 balls as he went 290 feet. Since no groups had used a proportional reasoning strategy and this was not directly connected to my learning goal, this strategy was not discussed. If students do not get the correct number of balls, it can be discussed what could affect this including that the gaps were not set up with the same length or that a better estimate of the gaps could be used. ReferencesBill, Victoria, and Margaret S. Smith. “Characteristics of Assessing and Advancing Questions.” University of Pittsburgh: Institute for Learning, 2008. Dude Perfect. 2018. “World Record Exercise Ball Surfing.” https://www.youtube.com/watch?v=i25i6vyjIpg&t=146s
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## Micah StohlmannChristian, author, and professor of mathematics education. ## Archives
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