Equations are Enlightening not Exasperating

According to numerous websites, the longest name belongs to a man named Adolph Blaine Charles Daivid Earl Frederick Gerald Hubert Irvin John Kenneth Lloyd Martin Nero Oliver Paul Quiney Randolph Sherman Uncas William Xerxes Yancy Zeus Wolfeschlegelsteinhausenbergerdorft Senior.  

Another man claims the record with Derek Parlane Stein Jackson Hunter McCloy Kennedy Scott Forsyth Henderson Boyd Robertson OHara Johnstone Miller Dawson Armour McDougall McLean McKean Fyfe McDonald Jardine Young Morris Denny Hamilton Watson Greig Wallace McQueen.

Yet another man wanted the title and changed his name to Captain Fantastic Faster Than Superman Spiderman Batman Wolverine Hulk And The Flash Combined- with 81 letters.

If these men were on a team together and wanted to put their names on the back of their shirts they would need a very big shirt or very small letters.  Suppose that they found a company willing to work with small letters to fit those long names on the shirts.

If that company charges a set rate for the shirt, say $15.00, and an adjustable rate for each letter, say .05¢, then what would be the cost for each person?  What would your cost be?   Figure it out with a formula.  Use "C" for cost, and x for the number of letters.

C = $15 + .05 x

Here is a fun and silly math activity:
Cut out little t-shirts out of construction paper.  (Here is another template with lines for writing.) Then have children come up with their own long name.  They could combine all of their pets, friends, family, etc.  When they are done, try out the formula above to find out the "cost" of each t-shirt.

Having fun and finding creative ways to let children try math and equations will give them the opportunity to be enlightened, not exasperated.

A Van, Volcano, and Vault...All Hold Volume

What is volume?  Let's explore and find out! It is how much is inside of something.  Everything inside the vault would be the volume.  That includes all the money, jewels, and air.

Figuring out just how much volume something has is done with a formula.
For the vault, a rectangular solid, the formula is:

      Volume = base x width x height or V= b x w x h

If you are planning to build a vault to put all of your treasures in, you want to make sure it is large enough to accommodate your future treasures, as well.
Another formula for volume you may use would be for a pool.  If you have the choice to purchase one of two pools, both the same price, other factors may help you decide.  If one takes less water, then it will use less chemicals to maintain it, too.  Use this formula to find the volume of these two pools:

     Volume = π x radius² x height or V= π r² h
     (Remember to convert feet to inches for both problems.)

Pool 1: 18 feet x 54 inches

21 feet x 52 inches

If you want to try a fun website that allows you to drag a dot to choose your own dimensions for a cylinder, then click here.

Help children understand volume by a visual experiment.
Gather:
     a plastic or glass container you can see through
     picture of water
     rocks, marbles, sand, and/or water proof toys of varying sizes
Begin by filling the empty see-through container with the rocks, etc., starting with the biggest objects first.  Next fill the container to the top with water.
Then ask the children if they think it would make a difference if they changed the order of how items were put in.   Have the children help reload the container starting with the water.  They may have figured this one out before you even begin-nothing else will fit. 
The ideal way to do this project:
     Have 2 containers, both see-through.  Fill one in this order; large rocks, small rocks, sand, then water.  Fill the other container and start with the sand, then small rocks, then large rocks, then water.  Less of the larger rocks will be used.  The idea is that children will understand how volume can be made up of different things, but the container still holds the same amount of volume.



   

Math and Art Collide with Tessellations

I love art, and math has one such art form that draws my attention.  It is called tessellation.  The patterns tessellation can create are most recognizable in quilts, but there are many ways to use them in real life.  Another recognizable use of tessellations is in floor tiles.  Below is an example of a beautiful quilt.

Basic tessellations are designed with a few basic rules:
1.  Each piece must be a regular polygon.
2.  Each vertex must be the same.
(Remember, this is a basic design, not a technical one.)
3.  The total of the vertex points must total 360 degrees exactly.
Squares work.  There are many shapes that tessellate alone or in combination with other shapes.

Print out your own sheet and create a fun color combination.  Here, see how triangles and hexagons work together to create tessellations?

My oldest son has a book about the Dutch artist, M.C. Escher.  His pieces of this art are so amazing and mesmerizing.  Here is one of them:

This is called "Horseman"

Another is entitled "Lizard":

If you would like to watch a video on how to create your own tessellation in the style of Escher, check out this video:

Polygons Please!

Polygons are exciting shapes to work with.  They offer so much variety, that they are never boring.  Two points that are connected is the beginning of a polygons.  Add enough line segments to enclose a shape and you have a polygons.
Polygons are named according to the number of sides.  My kids have fun with listing all the names they can remember.  Three line segments are needed to create the polygons with the least number of sides.  This would be a triangle.  Here is a chart to help you out:

Name	Sides	Shape	Interior Angle
Triangle (or Trigon)	3		60°
Quadrilateral (or Tetragon)	4		90°
Pentagon	5		108°
Hexagon	6		120°
Heptagon (or Septagon)	7		128.571°
Octagon	8		135°
Nonagon (or Enneagon)	9		140°
Decagon	10		144°
Hendecagon (or Undecagon)	11		147.273°
Dodecagon	12		150°
Triskaidecagon	13		152.308°
Tetrakaidecagon	14		154.286°
Pentadecagon	15		156°
Hexakaidecagon	16		157.5°
Heptadecagon	17		158.824°
Octakaidecagon	18		160°
Enneadecagon	19		161.053°
Icosagon	20		162°
Triacontagon	30		168°
Tetracontagon	40		171°
Pentacontagon	50		172.8°
Hexacontagon	60		174°
Heptacontagon	70		174.857°
Octacontagon	80		175.5°
Enneacontagon	90		176°
Hectagon	100		176.4°
Chiliagon	1,000		179.64°
Myriagon	10,000		179.964°
Megagon	1,000,000		~180°
Googolgon	10100		~180°
n-gon	n		(n-2) × 180° / n

For polygons with 13 or more sides, it is OK (and easier) to write "13-gon", "14-gon" ... "100-gon", etc.

Now for the fun activity to create polygons:
                                                        Sandbox/Beach/Snow Polygons
Supplies:
Protractors
Straight Edge (a yard stick or ruler)
Print out of the chart above

Head to the sand or snow to create your polygons.  Depending on the age of the children, you may or may not need to assist them.  Have each child draw as many polygons as they can. 

Remember to take pictures of this family math lesson.

When everyone has had some practice drawing polygons, see what creations can be made by drawing combinations of new polygons.  A few ideas to get you started could be a silly car, strange animals, a Farris wheel, flower, etc.
Another idea is to create a progressive drawing in the sand or snow.  One person draws a polygons of their choice.  The next person adds something, and so on until the group feels it is done. 


Here is a class video about polygons:

For snowflake polygons, check out this page:  Polygon Snowflakes
For a fun inside project, check out this page:  Star Polygon Wrappings

Lego Pizza Combination Math Activity

Legos are an amazing tool for children.  In this fun activity children get to create pizzas.

Yours will be a little different.

     Here is what you will need:
               "the crust"
                   - 15 paper plates (Shhhhhh...this is the answer, so just keep them stacked for now!)
          1 handful of each of these small Legos:
               "the meats"
                   - red (pepperoni)
                   - brown (sausage)
                   - gray (anchovies)
               "the vegetables"
                   - black (black olives)
                   - green (green olives)
                    -white (onions)

Now that you have your supplies, let's get started!  Grab the aprons, and if you don't have any, pin a towel around each child's waist.
At a table, explain that you are going to talk about combinations.  With pizzas, how many combinations of 2-topping pizzas can be made with six ingredients?  Let's find out!

When figuring combinations, remember that order does not matter.
The formula is:
                            nCr = ___n!___  =  _P(n,r)_
                                       (n-r) !r!            r!
Check your scientific calculator for directions on how to input this information correctly.
This activity will be hands-on, rather than paper and pencil.

Have your "ingredients" in bowls on the table.
Let the children decide what the first pizza will be, remember each pizza will get 2 toppings-no more, no less. 
Create one pizza at a time, using two ingredients for each.  Place a few (about 5) Lego toppings on the plate.
When the children are finished creating pizzas, there will be 15 different ways to make a 2 topping pizza from the Lego "ingredients".

Guide students as they work.  It may be best to start with one meat and complete all of those possible combinations, then move on to the next meat, then to vegetables.  Be careful not to repeat any of the combinations.

Here is a guide to check your results:
p = peperoni          bo = black olives
s = sausage            go = green olives
a = anchovies         o = onions

p, s          s, a          a, bo          bo, go          go, o
p, a          s, bo        a, go          bo, o
p, bo        s, go        a, o
p, go        s, o
p, o

All this talk of pizza may make you hungry for the real thing.  Here is my easy way to make a home-made pizza...not completely from scratch, but oh so good!

Ingredients for 1 pizza:
     1 Betty Crocker pizza crust mix
     1/2 cup corn meal
     5/8 cup hot tap water (=1/2 +1/8 cups)
     1/2 jar pizza sauce (pour the other 1/2 in a freezer bag, then in the freezer for your next pizza)
     1 cup sliced mushrooms (canned or fresh)
     1 cup sliced turkey peperoni (about 1/2 of a package)
     2 cups mozzarella cheese
Directions:
     Preheat oven to 400 degrees.  Mix crust mix, corn meal, and water together and set aside for 5 minutes.  Meanwhile grease a pizza pan and assemble ingredients.   Spread pizza crust with finger tips-this is a bit tedious, so start in the middle and continue to work out to the edge of the pan.  I like to use a greased pastry roller instead.  Then spread sauce to edge of crust.  Add your mushrooms and peperoni.  Top with cheese and bake for about 15-17 minutes at 400 degrees in the middle of the oven.

    My favorite way to change this is to add 1 Tablespoon Italian seasoning to the crust mixture, saute a package of sliced, fresh mushrooms with a large, sliced onion and a tablespoon of olive oil over medium-low heat until tender, then change the cheese to an Italian blend.  Otherwise keep everything the same as above.  Experiment with the ingredients; see what you and your family likes best.
    

Proportional Reasoning-Cooking Fun

Math and cooking go hand-in-hand...well, math is everywhere.  Let's have some fun making something to eat.  All the while, we will be learning about proportions.  What would cookies taste like if you put 1 cup of salt in instead of 1 teaspoon?  YUCK!  How about pancakes with 3 cups of sugar instead of 3 tablespoons?  Some kids might like that, but most people would think those pancakes would be too sweet.
*How much of one thing compared to another is proportion.  In preparing foods, it is very important to have the correct proportion of the ingredients being used.  Below is a recipe with proper proportions and alternate recipe to change the proportions and still have the cookies taste good.

Here we go!  We are going to make "No-Bake Oatmeal Cookies".   The original recipe came from my grandmother, who lived through the depression.

Mix these ingredients in a large sauce pan:
2 cups granulated sugar
1/2 cup milk
1/4 c. butter
Bring just to a boil and remove from heat.

In a mixing bowl combine:
3 cups dry oatmeal
1/2 cup coconut
3 tablespoons cocoa
1 cup chocolate chips

Add oatmeal mixture to sugar mixture and stir quickly.  Drop by rounded teaspoons onto wax paper.  Let cool and eat.

Let's change some proportions in this recipe.

What about butterscotch chips instead of chocolate?
     Keeping the proportions the same as they were, how many butterscotch chips should we use?

1 cup  That is correct!  We used one cup of chocolate chips in the original recipe, so if we are keeping the proportion the same then we still need one cup.

What if we don't want any coconut?  Can we just take it out?

No.  The mixture would be too runny and make a mess when we try to scoop it onto the wax paper.

What could we add more of to replace the coconut?

I was thinking oatmeal.  Is that what you were thinking?

Hmmm...how about making the cookie more caramel flavored?  To do that we can change half of the granulated sugar to brown sugar.  Our original recipe calls for 2 cups of granulated sugar.  How much of that should we change to brown sugar?

1 cup.  That's right!

Alright, now you found out that you will be having company.  You want to make more than you did for the last batch of cookies.  When you check your oatmeal, you find that you only have 4 1/2 cups of oatmeal left.  If you want to keep the proportions the same as the original recipe, how much of the other ingredients would you need?

3 cups of sugar

3/4 cup milk

3/8 cup butter

4 1/2 cups oatmeal

3/4 cup coconut

3 tablespoons cocoa

1 cup chocolate chips

Proportions are fun!  Enjoy eating your correctly proportioned cookies!
For additional math recipes check out this website:  Cooking with children kindergarten-12 grade