Using Math to Describe Gravity (from Playing With Math)

We are nearing completion of the book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. Our copy edit process was really a deeper editing process, and took over a year, with us working through a few chapters a week at first, getting everything just right. We finished it on May 9, Mother's Day. (My son was born on Mother's Day. I didn't realize until this moment how much I like it that our copy edit finished on Mother's Day too!)

One of the last pieces to go through copy edit was Sean's catapult activity that goes with Kate Nowak's chapter, Better Teaching Through Blogging. I had to build a catapult to test out the instructions Sean gave. I am not a crafts sort of person, so that had to wait until I had plenty of time to deal with it. Last fall, after I finally made my catapult, made my small adjustments to Sean's piece, and sent it to our copy editor, she  asked a lot of questions about the math. I began to realize that we needed an explanation of the math behind the catapult project.

So I wrote one last piece for the book, Using Math to Describe Gravity. I had fun writing this one. Most of what I wrote for the book took me a long time (and lots of agonizing) to write. This one was easy and quick. I finally realized that I really enjoy writing explanations. (I think I know my next book project...) I thought this might be useful to have online, so I'm including a modified version here. Enjoy!



Using Math to Describe Gravity

flickr.com/photos/joemacjr/189254474

In the picture above, we see water shooting upward. Whenever you squirt water from a hose pointed up at an angle, it follows a similar path. Have you ever wondered why? Math is helpful whenever we want to think about how something is changing.

For example, the idea of velocity tells us how position is changing with respect to time. (Unlike speed, which is just a positive number, velocity has direction, and can be negative to indicate a downward direction.) If you are driving at 70 miles per hour, in one hour you will have driven 70 miles. In two hours, 140 miles. Mathematicians generalize this idea by writing distance = rate * time.

The path of the water in the fountain makes the shape of a parabola. The physics of gravity explains why the water follows that path. As the force of gravity pulls us toward the center of the Earth, it creates an acceleration – a change in velocity. When we are near the surface of the Earth, that acceleration is always 32 feet per second squared (downward). That’s a weird unit, isn’t it? It means that if you are headed straight down, your velocity will increase by 32 feet per second each second. So when you drop something, one second later it has already gone from a speed of 0 feet per second to a speed of 32 feet per second. Over 20 miles per hour! In metric units, that would be 9.8 meters per second squared. (We’ll mostly stick with metric from here on out.)

This acceleration affects the relationship between distance and time because the speed, or rate, is changing. In the driving example, if your speed goes from 55 miles per hour to 60 miles per hour to 70 miles per hour, it makes it harder to calculate how far you have driven. (Calculus is great for understanding situations where your rate of change is changing, but I've written this for people who aren't familiar with the concepts of calculus, sticking to algebraic ideas.)

When considering the physics of situations like the fountain, we can analyze the vertical and horizontal motion separately. Gravity isn’t affecting the horizontal motion, so that stays constant. (If you were moving very fast, air resistance would slow you down. But at these speeds, we can ignore the effect of air resistance.) When you throw something upward at an angle, gravity pulls straight down, changing the vertical component of the velocity. Since the horizontal part of the motion is constant, this gradually changes the direction the object is headed, making the parabolic path you see above.

But how do we know that it’s in exactly the shape of a parabola? To see why it is, we’ll start with a simpler experiment, throwing a ball directly up. Now the path is no longer a parabola, because the horizontal position is not changing. But, amazingly, if we were to draw a graph of height versus time, that graph would still be a parabola. To describe parabolas algebraically, we use equations like y = at²+bt+c. In this case, y is the height and t is the time.

[Note: Depending on the flavor we want, we say the same thing in lots of different ways. rate * time = distance, R*T=D, velocity * time = height, h=v*.]

We can figure out a lot about a, b, and c by using what we know about the physical situation. When t=0, y = a*0²+b*0+c = c, so c can be filled in by knowing your initial height. If we had no gravity (and no air resistance), what we threw upward would keep going up with a constant speed. So its height would be given by rate (velocity) times time plus initial height. Do you see why b is the initial velocity the ball has as it leaves your hand?

That leaves a. The value of a will always be half the value of the gravitational constant - hmm, why half? In one second, the acceleration of gravity would increase velocity from 0 to 9.8 meters per second. So the average velocity during that second is 4.9 meters per second. In t seconds, we would increase from 0 to 9.8t meters per second, with an average of 4.9t meters per second times t seconds, for a height change of 4.9t² meters.

Now we can see the effects of initial height, initial velocity, and gravity combining to make an equation of the form y = at²+bt+c for height versus time. Gravity will actually affect an object in this way no matter which direction it’s pointed. And since the horizontal motion is constant, this same sort of relationship holds when we look at height versus horizontal position, although the values for a, b, and c will change.

When an object is launched at an angle, the value for a is determined by both gravity and the launch angle, b is determined by both the initial speed of launch and the launch angle, and c is still the initial height. To find the values for a and b, we can use the symmetry of parabolas across their vertex. If we can find the coordinates for the position of the vertex, we can use that to help use find the values for a and b.

All this thinking can help us understand the catapult activity better:

• When launching from the floor, where beginning and ending heights are the same, the x-coordinate of the vertex is just half the distance. So the vertex will be reached halfway through the time in the air.
• From the time the projectile reaches the vertex until the time it hits the floor, its height is decreasing at the same rate as an object that has been dropped, so you just use 4.9t² to find how far it dropped, which tells you how high it was.
• The vertex form for the equation of a parabola is y = a(x - h)² + k, where (h,k) represents the vertex, which we just found. If we assume that we launched from the origin, plugging zero in for x and y allows us to find the value for a.
• With the values for a, h, and k filled in, we have an equation in x and y. We can simplify it (change it to the form y = ax²+bx+c), and then modify it for raised launches by simply changing the value of c from 0 to the height of the launch surface. 
• We can use the new equation to figure out where to put a target we want to hit! The target will be on the ground, where the height is 0, so we can plug in y=0, and find x. In real life, the numbers that show up are almost never simple enough for factoring to work, so we’d need the quadratic formula. Measuring the horizontal distance from the floor right below the catapult, to the positive x-value from the quadratic formula, and centering the target there, we should be able to hit it. Candy bombing, here we come! 

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My thanks to John Golden for his help improving this.

Linkfest for Friday, May 2

Sometime in the past week, I added mathblogging.org to my feedly feed. They compile posts from all the math blogs they know of, with an interesting mechanism that takes you right to the original blog, instead of just linking to it. Yikes! I was already following hundreds of math blogs, and suddenly my feed doubled or tripled. I'll have to let it go eventually, it's just too much. But I found lots of interesting posts on blogs I'd never seen before, so today's linkfest will be more diverse than usual.

• MAA Minute Math (which has a new problem each weekday) had this problem on Monday: The volume (V), surface area (S), and total edge length (L) for a cube are consecutive terms in an arithmetic sequence. The largest possible side length is a+sqrt b (a,b integers). Find a+b.
• NUM+BER=PLAY, quite a bit harder than the ones that repeat letters/digits.
• 930 impossible figures.
• 37 pounds of garbage (kindergarten math)
• Rebecka's students loved their matrix-encoded scavenger hunt.
• Ashli is reading lots of interesting research. "The policy of requiring a passing score on an Algebra II exam for high school graduation simply cannot be justified."  
• Math for Writers is free for download, today and tomorrow.
• The tides of the quadratic formula. (What can you do with this?)
• Math and quilting, geogebra fun with John Golden.
• The current issue of Nautilus (an amazing online journal) is on symmetry. This issue includes a chapter a week for five weeks. This week's chapter is called 'In Theory', and includes articles on Penrose tiles and fair sharing.
• A primer on Newton's Laws, that starts out by emphasizing how counter-intuitive they are. (Physics, not math, but physics is the reason for calculus, right?)
• Megan Schmidt loved how engaged her students were with Nrich's Digit Doozy problem. I also like their largest product problem. I think I'll put a bunch of their posters up in the math lab in August.
• Bon Crowder says the "summer slide" is a myth. I concur. Let's play with math this summer.
• A puzzle posed as a card game.
• Tantalizing intro to 'The mirror crack'd: why physics is lopsided'. I'd like to read this, but I don't have a subscription. Bummer.
• Graphing game from John Golden.
• Why do math? from Siam, describes how math solves problems. Includes a description of google page rank (though it's not very in-depth).
• A and a half. How an Oregon Trail simulation demonstrates the problems with standardized tests.
• Engaging students in learning the multiplication facts, up to 20 times.

Crazy, how much good stuff there is to read. How can anyone absorb all this?

Caption Contest - Sometimes Learning Math Is Like Reaching Into a Hurricane

The book I've been putting together for the past 5 1/2 years, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, is almost done. We have chapters from over 30 authors, each chapter followed by a game, puzzle, or activity. It's looking great.

We are finishing up the illustrations now, and I need your help with one illustration. It illustrates the idea which informs the title of a chapter - At the Eye of the Hurricane, by Melanie Hayes. The illustration has two panels, sort of like a comic strip. And it reminds me of those collections of photos, with captions saying what different groups think you do. (Like this homeschooling one: http://www.lisaoutloud.net/Websites/lisaoutloud/images/homeschool.jpg)


Here's the text it illustrates:

We usually think of mathematics as a series of steps, starting with the foundational building blocks and eventually building a stairway to higher mathematics. We don’t move students up the stairway until they have mastered each previous step. We feel compelled to make sure they thoroughly understand algebra before we allow them to try trigonometry or calculus. For mathematically-gifted children, this lock-step method can kill their creativity and their desire to fit the pieces of the overall mathematical puzzle together.

The learning style of some mathematically-gifted children* is more akin to a hurricane; they stand at the eye and watch all the information swirling around them. Their curiosity urges them to reach into the hurricane and pull out bits and pieces of mathematical data. They ponder and experiment until they fit those bits and pieces into their prior knowledge and come up with the whole picture. To the casual observer or bewildered teacher it often seems disjointed and messy, but wonderful things are happening within the eye of the hurricane. These children are making deep connections, internalizing knowledge, and building concepts that will allow them to experiment and try out their own theories. Teaching mathematically-gifted children requires an open mind and a willingness to throw out most accepted notions of how to teach math.

And here's what we have so far...  (Thank you, Linda Palter!)

 

It needs a title and two captions. I think the title is just 'learning math', but maybe you have a better idea. The top panel could be 'what people think' and the bottom one, 'reality'. But that doesn't quite work. What do you think?


Prize: A signed copy of the book.



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* Although the author identifies this as a trait of gifted children, I think it is likely to be a good description for anyone who loves math. (The captions do not have to refer to gifted kids.)

Linkfest for Saturday, April 26

Partly I'm compiling these links for my own benefit, but some of this week's posts were pretty exciting. I hope you're out there enjoying them. Please let me know. Thanks.

• Varied, interleaved learning feels less effective, but is shown to be significantly more effective.
• Visualizing solutions to n-th degree algebraic equations using right-angle geometric paths. (I want to try to read this when I have more time and mental energy.)
• Mr. K describes how substitute teaching has made him a better teacher.
• Yakov Sinai models motion on a square pool table with a round cushion in the middle. He's proved that the ball will eventually hit every point on the table. This helps physicists understand gasses.
• Fawn's shopping contest - a way to practice percents (for discounts and tax) in a more challenging context.
• Nix the Tricks is a free ebook created by Tina Cardone with the help of dozens of math teachers online. Which 'tricks' are you attached to? Which make you cringe when your students mention them?
• A number of bloggers are answering some questions Grant Wiggins posed. Number one, why is division by zero undefined?
• Cathy O'Neill interviewed Fawn Nguyen on the math Common Core standards. Fawn is an optimist. I fear that the goodness she describes in the standards may be overpowered by the harm done with the endless testing, money for testing (instead of for teaching and learning), evaluations of schools and teachers determined by testing, etc.
• A game. Make a graph with the given number of edges at each node.
• Geoff's series of posts (especially number three on summer school with Jo Boaler, and number four on Chirstopher Danielson's hexagons) about the workshops he attended at the NCTM conference in New Orleans made me wish I could have been there.
• What do proofs with 5-year-olds look like? Rodi tells them, “I hear with my eyes and see with my ears.” And they work very hard to prove her wrong.
• I am dedicating Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers to Seymour Papert.  You can get some nuggets of his by following The Daily Papert. From Wednesday: “You learn in the deepest way when something happens that makes you fall in love with a particular piece of knowledge. “ 
• When students make their own definitions...
• Mr, Honner, Solving Pallet Equations.
• I have not managed to solve the Robot Maze Puzzle. Can you?
• Leading into calculus, estimating areas.
• Parametric projectile motion simulated in Desmos.
• Are you required to have the standard you are teaching on the board? Mr. K describes why that sucks. Spoilers, anyone? And then Brooke Powers describes how helpful it is to share your bigger goals at the beginning of a unit. (I don't think my way of doing that would satisfy the folks hovering over high school teachers. Thank goodness I teach college.)

This one's not math, but it's too good not to mention: Researchers are analyzing similarities between the behavoir of ants and neurons.

Linkfest for Sunday, April 20 (a small one)

• Nat Banting on making practice more conceptual - ask students to do the last step in posing the problem. Nice!
• Andrew Knauft descrbies why he thinks Geogebra > Desmos.
• A site for finding, building, and storing formulas online, Formula Sheet. (hat tip to Glenn Waddell, whose diigo account may have inspired me to get one - which I don't use. Maybe I should ask him to teach me how to make it useful. I love his real posts, but his Diigo Links (Weekly) are often full of useful ideas too.)
• Malke wonders whether lack of recess (and the movement it encourages) is taking away children's ability to make sense with their bodies.

Linkfest for Friday, April 18

• Jen Orr's first graders tell number stories and make a video of it.
• David Cox's post, Fostering the Hypothesis Wrecking Mindset, includes a list of a dozen good problems.
• Dan Finkel has been enjoying how much thinking goes on when people play his Horseshoes game, a variation on a classic number game. (Winner is the one who gets closest to the target number, using 4 given numbers.)
• Got some interesting data in a pdf file, and want to put it into a spreadsheet? Flowing Data points to a nice program to help with that.
• The 109th Carnival of Mathematics points to a game the author seems to like as well as 2048. I'll try it - Lunar Lockout.
• Toomai made a video - All Parabolas are the Same Shape.
• Mimi's Parametric Equations Animation Project
• Mr. K is introducing exponent properties with geometric series. I think I'm going to try to involve series in as many pre-calc topics as possible next semester.
• Jonathan Claydon wrote A Year with Desmos.

Linkfest for Saturday, April 5

• This video shows multiplying by using a parabola. Completely impractical, but I was curious why it worked. I figured it out and then wondered if my pre-calculus students could figure it out too. I wanted a demo instead of a video, so I built something in Desmos. (Hide the equations, and click on the three dots. The middle dot will always multiply the absolute values of the other two.) It's not perfect, but it might be good enough to impress my students.
• I've seen this cute list of functions, with the person's arms illustrating the graph, on a number of blogs lately. I see two that are wrong. Henri sees one wrong, and has quibbles with four of them. What do you see?
• Common Core for math... I keep hearing that the math standards are pretty good. But if the tests ignore the most important standards (the process standards, which describe mathematical thinking), then they're being used badly. This post by Jonathan Katz goes into some detail.
• Nice exercise. One person looks at the board, and describes the graph drawn there. Their partner must draw it from the verbal description.
• Quintic polynomials. There is no formula for the roots. But there is this. I want to learn more!
• Fawn's lesson for proportional thinking.
• Papert on "hard fun."
• I like this diagonal problem, but when I tried it in class my students were not persistent enough to succeed with it. David Cox's post on how he used it with his students makes me want to try it again.
• In whatif?, xkcd's creator, Randall Munroe, takes a silly question and analyzes it with math and physics to come up with an answer. In this episode, he figure how how big a splash you'd get from a tree as big as all trees on earth falling into an ocean with the water of all the ocean's on earth.
• In this post from her calculus for kids series, I like Maria's thoughts on how we help kids learn problem-solving.

Linkfest for Sunday, March 30

• Michael Pershan asks some good questions about how to approach the teaching of complex numbers.
• Fawn Nguyen has another Math Talk with her students. (I'm wondering if students think about which sorts of visual patterns will give a quadratic function and which will give a linear function.)
• OpenMathBook, a blog "to promote, discuss, and develop free and open source mathematics textbooks".
• Ihor describes a high school class recreating the experiment Eratosthenes conducted 2200 years ago. (Some day I want to try it...)
• Maria is conducting calculus math circles with 7 to 11 year olds. I am fascinated. My understanding of calculus is so steeped in algebraic thinking, it was hard for me to imagine at first. I love what she's doing.

Guest Post: John Spencer Addresses "Frustrated Parent"

John's was the first post I saw about the silly complaint going around from "Frustrated Parent". (Now I've seen about three more. They all have good things to say.) John has graciously allowed me to share his whole post here. (But the comments over at his place are an interesting mix, so go on over there too.) Here's John:

There are many things I hate about the Common Core standards. I hate the way teachers were pushed out of the creation and adoption phase and how we have little voice in the implementation. I hate the fact that the standards will continue to be assessed with standardized, multiple choice tests and that these scores will be used with Value Added Measures in both teacher salary and teacher evaluation. However, I think it's important that in our criticism of bad policy we are careful to avoid blasting good pedagogy.


I'm seeing many of these posts making their rounds on Facebook.

I'm seeing statements like, "What the hell is a number line and why do kids need it?" Or, "just teach them the basics." The notion of using a manipulative, playing with numbers, breaking them them apart and comparing processes is somehow viewed as non-mathematical.

The truth is that number lines are powerful tools for understanding integers. True, when subtraction is something simple that requires no "borrowing" it feels like a joke. However, the goal is to build up number sense. It's to help them understand math conceptually. If you flip the numbers and end with a negative number as an answer, suddenly a number line helps make the negative-positive relationship more powerful.

This parent's snarky answer about "the process used would get you terminated" is based on a faulty assumption that a first grader needs the same approach as an engineer. And yet . . . this "new math" approach that people mock is something we use constantly in real-world, mental math.

Consider it this way: You have fifty-three dollars and you need to give someone twenty-seven dollars. What are you going to do to figure it out? If you find yourself breaking by tens and going backward, chances are you are using a mental number line.

Oh, you could pull out a piece of paper and do that math that way, but chances is are that as an engineer, you'd be fired . . . or at least laughed at.

I remember someone posting an angry rant about doing multiplication by breaking it up into different pieces. "Just teach the algorithm!" the parent posted.

I posted a response. "If the bill is 27.42 and you want to leave a twenty percent tip, what's the answer? How did you find it?"

Some people divided by five. Others multiplied by .2. Still others moved one decimal over and doubled it. Some rounded up to thirty. In other words, there were multiple processes that worked and each of them involved understanding the properties of numbers. In other words, most people used a process mentally that they were openly mocking on Facebook.

*   *   *

Oddly enough, many of these same people who are mocking "new math" in their posts are also lamenting the fact that Singapore is kicking our butts in math. What they fail to realize is that the places where math is working are the places where they are building number sense.

I've seen what happens when students lack number sense. They learn a lockstep process and think that math is the same as baking a cake. They follow the recipe without understanding why they are doing what they are doing. However, when they get into something as simple as linear equations, they struggle to know what to "do first," when there are often two or three options.

When students lack number sense and they get the wrong answer, they fail to understand why an answer is illogical. You end up with a student who misplaces a decimal number and never finds his or her mistake. Asking students to think conceptually and engage in diagnostic problem-solving isn't superfluous. It's actually part of "the basics."

I know that the "new" math looks different, but instead of criticizing it for being hard or being complicated, try thinking about the theories behind it. There's a reason we're using manipulatives, breaking things apart, using number lines and comparing processes. This is how math works.

Linkfest for Thursday, March 27

• I got my master's degree at Eastern Michigan University. It makes me happy to see them on the web.
• Colorful related rates problems, and colorful descriptions of solution strategies.
• Turning multiplication table practice into a scrabble-like game sounds like a good idea to me.
• Teaching metacognition in math class.
• Fawn is teaching absolute value. (And giving us some great ideas.) And James Cleveland has a great inspiration, using estimation180 to help teach absolute value and analysis of errors.
• Math Munch collects other games like 2048.
• Mr.Honner once again finds a weird graph on a standardized test. It purports to be exponential, but it's not.
• Justin Lanier did a math circle on Celtic knots.
• Interesting puzzle - place the houses on the number line. I know there's some math here, but I haven't figured it out yet.
• Optimization, a calculus problem solved by dogs, ants, and light.
• Henri Picciotto has a nice visual Pythagorean Theorem proof.