People often learn math best with concrete, reallife, physical examples that they can feel, touch, move and manipulate. Here are a few ideas for handson physical math activities that bring life to more complex math concepts.
It IS Rocket Science!
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 everwonderful triangle to help you out. Butit 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 withwe 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 roundside 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 180168=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= 10/h
0.213 adj =10
h= 10/0.213
h= 46.948
Therefore the rocket reached a maximum altitude of just under 47m.
Tiling Pattern Challenges
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 (manysided 2dimensional geometric shapes) and their tiling patterns. Some of his patterns are in the pictures above.
Challenge #1:
 Choose a number between 3 and 16.
 Make an equilateral polygon (every side has the same length and all the angles are equal) with that number of sides on card stock or other stiff paper and cut it out.
 Repeat with 2 other different numbers.
 Trace the first shape onto another piece of paper then repeat such that the second tracing shares a side in common with the first. OR, trace and cut out several more of that shape so you can freely move them around.
 If you are tracing them, colour in the shapes each time you trace a new one. Can you repeat tracings such that there is no leftover empty space? Repeat with each of the polygons you have cut out.
If there is space left over, does it always take on the same shape? For example, an octagonal (8sided) shape with consistently leave squares in between.
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 nontiled spacedoes 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
Proportional Recipes
Lemonade Challenge:
Make the best lemonade with only three ingredients: lemon juice, water & sweetener (sugar or honey).
Materials:
5ml spoon
15 ml spoon
Graduated clear (metric) measuring cup
Pitcher of water
Lemon juice
Sweetener
Pencil and paper for recording recipes and results
Measure each trial carefully, and with a partner or two, determine the perfect proportions for your recipe. Try at least 3 different recipes to determine the one that is best.
The rules of sale vary from country to country. If you were to market this, you could find it necessary to express your recipe on the label in different ways. Rewrite your recipe as follows:
 Ingredients listed in order from the most to the least used per serving
 As a ratio
 As percentages (by volume or by weight)
 As fractions (by volume or by weight
Step it up:
Make a lemon iced tea recipe by adding tea as a fourth ingredient. You can use instant tea, or get creative with brewing your own (1 teabag per 500 ml of hot water, steeped for 10 minutes then remove the bag and refrigerate).
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 34 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).
Field Day
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 eitherthere are track and field markings, hopscotch, 4square, corn and hedge mazes, airport runways, lparking lots, andscaping patterns...there are many reasons why people find it necessary to mark out largescale geometric patterns. So, how do you do it?
First, decide what it is you need. Want a baseball diamond? Then you will need to be able to measure the length of the baselines, the angles from home plate and between the bases, the distance to the pitcher's mound, the batter's box, and the coaches boxes. For some fields, you will need to be able to draw circles or segments of a circle. For example there are circles on hockey arenas, and semicircles on basketball courts.
You will need a way to mark the lines. For an outdoor sports field, chalk is often used, and there are special dispensers you can use for this. For landscaping, you will likely use sring and stakes to mark out areas. The way you make the shapes is quite similar to how you would do it on paper, but the tools are somewhat larger.
How can you make a very long straight line?
How can you make a large circle or arc?
How can you accurately create specific angles on a large scale?
How do you ensure all your measurements line up properly?
Once you have wrestled with the answers to these questions and have decided on the answers, you are ready to begin.
Having trouble? This link and this link will help answer some of your questions, but to get the most value out of this exercise, only use these as a last resort! You can always do an internet search for marking playing areas for additional sports as desired.
If this "field" interests you, you may want to investigate the job of surveyor, civil engineer or landscaper who use these skills on a regular basis.
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.
My Infinity is Bigger Than Your Infinity
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/howmanykindsofinfinityarethere/ and shares some proofs here: https://www.youtube.com/watch?v=lA6hE7NFIK0
How many infinities can you find in your everyday life? I have my own example to get you started: adding water to a countertop water filter at a rate that approaches the rate of filtration, but is just slightly off. In fact, by just exploring limits as used in calculus you can find many, many (infinite?) examples.
Packing Problems: 3D Tetris for Real Life
Try out this practical reallife version of the popular game 3D Tetris where students are encouraged to literally think inside the box.
How many of us are challenged by spatial and volume challenges on a daily basis? This activity applies the mathematics of spatial geometry and volume to a very practical handson problem.
For this activity, you will need:
A large cardboard box
A variety of smaller containers or items of a variety of shapes and sizes such that a given variety does not easily pack perfectly into the larger box example: various sized cylinders, cone shapes, triangular prisms, rectangular prisms, spheres (basketball, volleyball, softball, grapefruit) etc.
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.
Ancient Mathematical Tools
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.
Sundialtime
Barometerweather
Sextantnavigation
Astrolabenavigation
Plumb linebuilding
Compassnavigation
Water clocktime keeping
Altimeteraltitude
Barometer
A barometer can help you detect changes in atmospheric air pressure and predict clear or stormy weather.
To make a barometer you will need:
 A metal can with the lid removed such that there is a protected (not sharp) edge left
 A round balloon
 A thin marker
 A small plastic stir stick
 A 2 cm long piece of sticky tape
 A ruler
 Cut the end off the balloon and stretch the rest over the opening of the can
 Mark a line all the way across the craft stick at the edge of the top of the can to line it up should it accidentally move out of place.
 Tape one end of the stir stick to the centre of the balloon.
 Place the bottom of the ruler on the same surface as the barometer so that the loose end of the stir stick points at the ruler and the "0" is at the bottom.
 Record the places on the ruler that the stir stick points at different levels of air pressure. When the air pressure is high, the balloon will lie flat or even sink down a little causing the loose end of the stir stick to point up; when it is low, the centre of the balloon will rise and the stir stick will angle down the "hill" made by the balloon.
Plumb Bob
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.
To make a simple plumb bob, you will need:
 A short pencil with an eraser attached
 A small screw with an eye on the end
 A piece of thin string
 Centre the screw on the eraser of the pencil and screw it in a few twists.
 Fasten a piece of string to the eye on the screw.
 Hang the other end of the string from the object you wish to measure plumb.
Make Your Own Magnetic Compass
You will need:
 A sewing needle
 A small container with enough water to float the needle upon
 Waxed paper, cut into a circle slightly smaller than the mouth of the container, or a small round piece of cork
A bar magnet
 Rub the needle against the magnet in the same direction repeatedly in order to magnetize the needle.
 Carefully lay the waxed paper on the surface of the water.
 Lay the needle on top of the waxed paper, or poke it through the cork until the cork sits in the centre, then float it on the water. It should turn until one end points north, and the other end points south. If the needle doesn't turn properly, continue stroking it along the magnet until it does. This action aligns the electrons in the needle thereby magnetizing it.
 Label the directions on the cup and you can line up the needle to determine direction.
Make Your Own Water Clock
You will need:
 2 or more containers you can punch a hole in (cans or plastic jars/ bottles, paper cups etc.)
 A small box, brick, book or other "riser" to place the top container upon
 Water
 Small drill, awl or sharp implement to make holes in the containers
 A watch or clock aside from the one you are building that has a timer and/or second hand or counter (for calibration)
 A thin permanent marker for marking levels and times
 Cut a small hole in the bottom of the side of one of the containers.
 Line up the hole so that it will pour into the second container. Do the same for the next container and so on if you are using more than 2 containers.
 Carefully pour water into the top container and immediately begin timing. Mark the level at the 20 second mark, 30 seconds etc. with the permanent marker. Repeat several times to ensure accuracy and to allow yourself time for labelling. This is called calibrating.
 Once it has been calibrated and labelled, it is ready for regular use.
The Mathematics of Slime Mould?
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.
For more on slime mould and its fascinating growth patterns, click here.
For more on Conway's Game of Life and how to play, click here.
More math games, manipulatives you can make and additional math activities can be found here.
Also check my blog for more math and other educational activities:
http://lemonadebyll.blogspot.ca/
More Math Links from Lemonade:
 Math Links
Spatial Mathematics:
Mobius Strips and other Dimensional Wonders: the Mathematics of Topology
Mobius Bagels and Other Topological Challenges
Graphically Simulated Video on Mobius Transformations
Mandelbrot Sets in 3D: Math and Art Meet in Chaos
Origami, Architecture and Topology Photography
Penrose Tiles
Geometrical Experiments in Art
Explorations:
Scratch: MIT's kid's programming language (free download)
Hilarious Videos on Mathematical Ideas from Vi Hart
Magic Squares, Fibonacci Series and other Fun Numbers to Explore
Excellent Infographic All About Pi
How Archimedes Approximated Pi
All About Quantum Mathematics
More Math and Strategy Games
Paper and Pencil Math Games
Wild About MathMore Paper and Pencil Games
Instructional, Applied Mathematics, Math Contests, Tools
The Myth of Being "Bad at Math"
Better Than PEDMAS: Why Having Students Memorize the Order of Operations Does Them a Disservice
Khan Academy: Online Instructional Math Videos
Easy Explanations for Advanced Math and Computing
Jim's Algebra Hints
Solving Equations as Proofs
The Best Way to Factor Trinomials
Engineering Megasite of Applied Math and Science Activities and Resources for Grades K12
CEMC: Home of the UW Mathematics Competitions (loads of math resources)
Canadian Math Contests
Wolfram Alpha Computational Search Engine
All About Sliderules and How to Use One
101 Uses for a Quadratic Equation
101 Uses for a Quadratic Equation Part 2
Resources for Math Teachers
The Myth of Being "Bad at Math"
Biographies of Mathematicians
Common Math Mistakes
National Library of Virtual Manipulatives
Illusions: NCTM Virtual Manipulatives
Geometer's Sketchpad (sorry, this one is not free)
The Power of Grace in Teaching
Math Books: Big Ideas for Small Mathematicians &
Big Ideas for Growing Mathematicians Ann Kajander
Math for Smarty Pants Marilyn Burns
Math Games for Middle School Mario George Salvadori
MindSharpening Logic Games Andrea Angiolino
Mathematics Made Simple, 6th Ed. Thomas Cusick
Chaos James Gleick
Also: look for math titles by Martin Gardnerthere are too many to list here
Other Resources:
Microcosms Brandon Broll (Microscopic images up to 20 million x magnificationI added this because it shows patterns, scale, Fibonacci in nature, etc.)
Online Magazines & General Math Sites
Plus Magazine
A+ Click
