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People often learn math best with concrete, real-life, physical examples that they can feel, touch, move and manipulate. Here are a few ideas for hands-on physical math activities that bring life to more complex math concepts.
First, familiarize yourself with the basics of model rocket safety.
To determine how high a rocket went, aka the maximum altitude, you can use the ever-wonderful triangle to help you out. But-it must be a right angled triangle for our purposes.
Trigonometry allows us to figure out all of the secrets of a given triangle with only three pieces of information. It doesn't matter which ones we start with-we could have the length of 3 sides and figure out the 3 angles from it, the 3 angles and figure out the side lengths from those, any 2 angles and a side to figure out the others, or any 2 side lengths and an angle to figure out the rest.
Looking at our situation, what do we know?
Well, angle "L" (for launch pad) is 90 degrees. We also know the distance we marked out along the ground. Let's call that side g (for ground). If we want to be extra accurate here, we will need to subtract the height of the person (up to their eyes) who will be at that point from the length of that line. We'll call that point V for view point.
Here's where we get tricky. We will need to use a protractor to measure out the angle from point V to the top of the rocket's trajectory. You can do this by taping a string to the 0 point on a protractor and tying a small weight (a washer works well) to the other end. This will make an altimeter. The viewer holds the protractor round-side down and looks up along the bottom side (the 180 degree line, or base) and points the other end up to where the rocket reaches maximum height. A second person reads and records the angle that is marked by the string on the protractor.
Say we read an angle of 78 degrees off of the protractor, we can add that to 90 degrees (our angle "L") to get 168 degrees. That tells us that the remaining angle (T) must be 12 degrees since 180-168=12.
OK, but I want the height, not all these angles!
This is where a little trigonometry comes in to play.
We know: Soh Cah Toa (if you don't, you can review this here).
We know the length of the bottom line, g, since we chose it by measuring directly. Let's say it is 10 m. We want to know the length of the line h. We'll call the point at the top of the rocket's flight point T (for top).
We can use tangent, since tan=opposite/adjacent
We have a choice of which angle we can use since we know the value of all three. For this example, I will use the angle at the top of the rocket's flight, which we'll call T. T=12 degrees.
So, tan 12= 10/adj since the opposite line, the line along the ground is 10m We have already labelled the adjacent line as h for height.
Using a scientific calculator, we find the value of tan 12 to be 0.2125565 For rounding purposes, we will use 0.213
0.213 adj =10
Therefore the rocket reached a maximum altitude of just under 47m.
Maybe you. like me, have found yourself waiting in a train or bus station or a place of worship that has a mosaic of floor and/or wall tiles. Some can be quite interesting, and I used to play mental "what if" games of making new patterns by changing one or more tiles in a series in my head. Some of the most interesting patterns involve a variety of shapes. Mathematician Roger Penrose studied polygons (many-sided 2-dimensional geometric shapes) and their tiling patterns. Some of his patterns are in the pictures above.
Without cutting out more shapes, can you predict whether a new polygon will have leftover space or not?
Challenge #2:For this challenge, you will need a very large tub of pattern blocks and a surface with a fixed area (desk or table top, marked off floor space, etc.). The challenge is to create a repeating pattern with the blocks that exactly fits the area you have been assigned. How close can you get? How about if the pattern doesn't need to repeat, but does need to be free of non-tiled space--does that improve your chances of success?
Repeating patterns of polygons with no overlapping or spaces are called tesselations. M.C. Escher was fond of using these in someof his work. His art is not on public domain, but you can follow this external link to see some of his work. While you are tere, also look at some of his other work of interest to math enthusiasts. Challenge #3: Play. Seriously! Use the tub of pattern blocks, and/or shapes you have made with the tacings above, and experiment. Can you create a mandala such as this one
Crazy Cookie Challenge
Using a common list of ingredients, divide a large group into smaller groups. Each group will develop their own proportions, bake a small batch of cookies (about 3-4 cookies worth), and compare with the other groups. Winning categories may include crunchiest, chewiest, tastiest, etc. Participants will then work out the proportions for baking 1 or 2 dozen cookies using the same proportions they used for their original batch. Hint: to make eggs easier to split into smaller mounts, try using powdered eggs instead of fresh ones for this activity).
Have you ever noticed how many sports require a very precisely marked playing field? From cricket to squash, to baseball and basketball, sports often rely on geometrical patterns for their play area. In fact, it isn't just limited to traditional team sports either--there are track and field markings, hopscotch, 4-square, corn and hedge mazes, airport runways, lparking lots, andscaping patterns...there are many reasons why people find it necessary to mark out large-scale geometric patterns. So, how do you do it?
Challenge 1:Invent your own game and mark out your own related playing area. Indoors, use masking tape on a floor; on turf use powdered chalk, and on asphalt or concrete, use sidewalk chalk. Play your game and teacher it to others, tweaking the rules and playing area as needed.
Challenge 2:Create a landscaping plan to grow a vegigie and/or flower garden. Don't limit yourself to rectangles and rows! Try using circles, triangles, hexagons, etc. to make it more interesting. Also, remember to leave a little space to walk to tend to and harvest your crop. Once you have uour design, build and plant it.
We have all heard about infinity, but did you know there are many different kinds of infinity? ViHart describes a few here: http://vihart.com/how-many-kinds-of-infinity-are-there/ and shares some proofs here: https://www.youtube.com/watch?v=lA6hE7NFIK0
Try out this practical real-life version of the popular game 3D Tetris where students are encouraged to literally think inside the box.
The challenge:Calculate the volume of the large box. Calculate the volume of each shape you are able to put into it and still close the flaps complete shut. Record each shape and its volume as you add it to the box. Rearrange (and adjust your list) and repeat as necessary in order to fit the maximum volume of containers and leave as little empty space in the bigger box as possible. Compare your results with other groups and express your results as either a ratio or percentage.
Be sure to stipulate that items cannot be opened, crushed or nested and must remain in the same condition they began.
From building pyramids, to navigating across vast bodies of water, people have used their ingenuity to create useful mathematical tools for thousands of years. Here are a few you can make yourself.
A barometer can help you detect changes in atmospheric air pressure and predict clear or stormy weather.
A plumb bob is a useful tool that enables you to find a line perpendicular to the ground by simply using gravity, a string and a weight.
You will need:
You will need:
Researchers have found that slime mould growth patterns can be useful tools for modeling other systems such as subway routes, communication networks and more. Slime mould has even been shown to be able to solve mazes! Similar patterns of growth have been modeled by Conway's Game of Life and the mathematical patterns involved in patterns such as these are providing insights into a wide range of possible applications.
More math games, manipulatives you can make and additional math activities can be found here.
More Math Links from Lemonade: