You are standing on the sidewalk. Somebody yells, “watch out!” And you look up and realize a television is hurtling towards your head. You have 2 seconds to move out of the way. What floor was the television dropped from?
This is a reverse of the classic egg drop problem, which asks the student to figure out how long it will take an object to fall from a specified height. The formula for gravity and time is usually provided. We are going to turn this lesson on it’s head, literally!
The lesson is designed for two 45 min lesson periods. I’ll give the standards and practices at the end (or maybe I’ll let you tell me which ones this lesson hits!)
The lesson starts with an exploration of gravity and a ‘where does the formula come from’, and moves to the exploration of the above scenario – quickly, before somebody gets smacked on the head by a TV! (The title of the blog is the question that kicks off the gravity part!)
Start with the question: So let’s say a television is falling on your head. Will a bigger TV kill you faster?
Have the students make conjectures in their own minds. Have the silent thumbs up when they are ready. Pair students up and have them share their conjectures, giving reasons that the other person can articulate. Go around the groups and ask what they decided: what the question was asking, whether a bigger tv would fall faster than a smaller one, give the reason(s) they felt that was true or false.
If your facility offers a one or two story drop (football stadium announcer stairs?) conduct this next experiment empirically.
Or, use this great video called Misconceptions about Falling Objects.”
First, watch only the first part of the video, where the interviewer is asking random people whether a heavier ball will fall faster or the same as a lighter weight ball.
Stop the video before they get to the proof part. Survey your students. See how many agree with the heavier is faster theory. Ask them why. Have them explain their thought process.
At this point, you can take them outside, let them try it themselves with two different weight items, or you can play the rest of the video, or both.
If you get to do the outside drop, bring a stopwatch. Set up some students to drop and some students to time the drop and determine how fast the ball drops. Have the students estimate the height (if you can, measure it ahead of time, so you can work the math first) and come back to the class.
The task ahead of the students is two- fold:
1. To understand that items fall at the same rate, no matter the weight, and to perhaps extend to the connected idea that gravity is a pull that has nothing to do with weight – weight has to do with density of matter. (If you have a wonderful science teacher, maybe she will work on a connected gravity lesson!)
2. Develop an understanding of the parts of the formula used in so many of their quadratic one-variable math problems, as they develop their speed of gravity based on the tests, or in discussion from the information from the video.
I will tell you that the video gives the formula in meters per second. The formula in the Alg II Word problems are generally given in feet per second so take some time here to work the students through a conversion process, so they will see the variations as different ways to write the same formula.
Rest or break here. Debrief the students, let them write out their understanding of the question asked in the title- will a bigger TV kill you faster? See how many students changed or enlarged their views. Have them articulate what changed, or if they thought objects would fall at the same rate, how did what they did support their initial thought.
Pick up the next day with the formulas. Let the students (whole group) come up with thoughts as to how they are different and why one formula might be more appropriate than another.
Now, pose the second scenario:
you are standing on the sidewalk beside a multi-story building. Above your head, someone has knocked a window air-conditioner out of the window. If you don’t move it will land on your head. You have 2 seconds to move. How high up does that window have to be to give you time to get out of the way? If you double the height, does that double the amount of time you have to move?
Have the students each think about the problem individually and observe their ideas. You may need to ask some questions of those students who can’t get started, to help them find an entry point. Ask them if they could draw the problem, or have them list things they will need to know (rate of gravity) to solve the problem. (Working the problem looking for height requires re-arranging the formula, which is why I wanted the students to really understand the parts and where the numbers come from.)
The next step is group discussion. In groups of 3 or 4, have them share their ideas and work together to discuss how they feel they can solve the problem, discuss which version of the formula will be appropriate to use, and details, like the average height of floors in a multi story building.
Encourage through questions some hypotheses, maybe a drop from the second floor. At this point there will be some discussion about how to mathematize the answer. One of the recurring problems for students deals with the square root solutions required to solve quadratics. Students should have already worked through the various ways to solve quadratics, so you will want to listen to the conversations surrounding these issues.
Some students will be able to solve for the square and some will not. Have the student who does solve the square explain his/her understanding of the process. If students in each group come up with differing ideas, let them question, critique, convince, until they can agree on some solution. After all, their life depends on knowing how fast they must move!
As you listen, decide which groups to call on for sequencing purposes. Bring the group together and let each group talk about what information they used to solve the height issue. As students speak, if someone realizes their mistake, let them correct it, then iterate the thought process – why did they think the mistake was reasonable?
Continue to ask questions to guide the students through the ideas, asking students to reiterate what they heard another student saying; asking a student to clarify a point, to dig deeper into the thought process.
They should, of course, come up with two answers, positive and negative. So the next question is which answer makes sense and why? What does the negative answer represent? What would have to happen for the negative answer to make sense?
The students can practice their skills by asking them the same problem, only transferring the location to Mars. If the television dropped from the same height on Mars, would they have more time or less?
Don’t forget to ask about doubling the distance. Let the students confirm whether or not that is true and why.
This is a good time to pull the students together and have students restate the ideas they’ve heard. Watch and listen for evidence of a change in student understanding. Are they making the same errors or different ones?
To cement the knowledge, have this similar problem ready: you are in a boat at the foot of a cliff. You look up and see a man tumble straight down. Develop some different scenarios for the height of the cliff and the time it takes him to fall.
Okay, I won’t make you guess the standards and practices- but if you see any I missed, let me know!
The CCSS task that is directly addressed here is Algebra II: Quadratic Equations in one variable (standard A.REI.4)
The activity supports all eight mathematical practice standards. The various individual, small group and whole group tasks give plenty of opportunity for formative assessment of math skill levels and understandings. The Mars thing – have either an internet access to look up the gravity on Mars, or have it ready and let the students who get done early work on the Mars formula for the rest of the class to use. They can do a mini present on how they got it.
There are sure to be multiple paths students took to get the answers. Let whole group discussion offer opportunities for students to share how they thought about solving. Put the paths on the board and let students compare the paths – they will find ways to add some of these new paths to their own toolbox if skills!
As always – feel free to use, modify, comment, and question. Thanks!