Measuring Noah’s Rainbow Arc

Have your class make a homemade Noah’s arc.  You will need creative materials:

a ruler

a tetra pack or other recycled container that floats

sticky pine pitch or an eco-friendly sealant

other thoughtful decorative creative materials

In the bible, Noah is instructed to make an arc large enough and strong enough to fit a lot of animals and to last in the flood that is to come.  The name Noah is noted as “comforter”:  Make thee an ark of gopher wood; rooms shalt thou make in the ark, and shalt pitch it within and without with pitch.  (Blue Letter Bible; Genesis 6:14)…And this is the fashion which thou shalt make it of: The length of the ark shall be three hundred cubits, the breadth of it fifty cubits, and the height of it thirty cubits.  (Blue Letter Bible Genesis 6:15.  A window shalt thou make to the ark, and in a cubit shalt thou finish it above; and the door of the ark shalt thou set in the side thereof; with lower, second, and third stories shalt thou make it.  (Genesis 6:16) (This passage could either be 4 stories in Height in its description, or 3, depending how it is interpreted – is the lower basement floor considered to be counted as a floor.  The passage in Genesis (Genesis 6:15) says that God instructed Noah to build the Arc in these dimensions using Cubits.  The cubit is an ancient unit based on the forearm length from the tip of the middle finger to the bottom of the elbow.  The estimate varies depending on which version of a biblical text one reads.  Approximately 17.5-20.6 inches (  What in Today’s world can be compared with The Length of Noah’s Arc about 450 Feet Long?  a Baseball Field; a 7 story Building.  There would be 3-4 stories of height (including the lower) and a giraffe would have to fit (approximately and up to 15-18 feet)!  How tall is a giraffe?

Cubits Answer:

300 Cubits = 450’ L

50 Cubits = 75 ‘ W

30 Cubits = 45 ‘ H

where L = Length

           W = Width

           H = Height

Metric Conversion (where 1 inch – 2.5 cm):

L   300mm = 30cm

W 50 mm = 5cm

H 30 mm = 3 cm

Have your class find homemade materials from the recycle bucket or pieces of materials that your folks have no need for, and make a miniature version of the arc as it is described.  Fashion a window 18 inches from the roof, and make a door.

Rainbow Covenant (Genesis 9:11-16… And I will establish my covenant with you; neither shall all flesh be cut off any more by the waters of a flood; neither shall there any more be a flood to destroy the earth….And God said, This is the token of the covenant which I make between me and you and every living creature that is with you, for perpetual generations:…And I shall set my bow in the cloud, and it shall be for a token of a covenant between me and the earth….


(Photo Here)

Our homemade prototype turned out to be 12 inches x 1 inch x 1.5inches, with a window just under the top, and it floats!  Have fun decorating your Arc as you would be living it for 150 days before the waters receded.  Pine Tar is a term for what is called “Pitch”.   It can act as a sealer for the bottom of your arc, but be careful as it is sticky stuff!  Have fun!

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Fractions in The Kitchen

Choose a recipe from home; notice the fractions used in the recipe.  Formulate fraction questions; and calculate the questions.  Then go ahead and have an baking extravaganza in your school’s kitchen, or at home.  Here is a sample:

Potato Tea Buns

This classroom kitchen recipe was a combination of Tea Buns from the Telephone Pioneers oF AmerIca, Ch. 49; Nova ScoTIa; WhaT Am I Gonna Cook? RecIpe of PaT Brooks; HunT’s PoInT, NS; with some of our personal add-ons such as Brown sugar

1 Pckg Dry YeasT

1/4 cup waTer Mixed wITh 1 Tsp Sugar sIc. We prefer Brown; HealThIer)

1/2 Cup Mashed PoTaToes

1/4 Cup BuTTer (sIc. PaT says ShorTenIng or MargarIne; We prefer Real BuTTer)

1/4 cup Sugar (sIc. We prefer Brown; or a bIT of Molasses, Though The rolls would be a dIfferenT Color.

1 & 1/2 Tsp SaLT

1 Cup Milk (sIc. Almond Milk)

1 Egg (We prefer a dollap of Flax Gel, made by parboIlIng 1 Tbls Flax Seeds In 1 Cup of waTer for 10-12 mInuTes)

4 Cups WhITe Flour

CombIne WaTer and 1 Tsp Sugar and YeasT

LeT sTand For 10 mInuTes

In a saucepan, combIne Milk, PoZTaToes, BuTTer, SaLT and Sugar;

heaT unTIl BuTTer has MelTed

Add yeasT mIxTure To Flax Gel In a Large BowL

STIr In boTh Cups of WaTer and BeaT well

Add RemaInIng Flour

Place In a Warm spoT for 1 Hour unTIl double

Cover wITh Damp CloTh; Leave For 1 Hr To RIse

Bake aT 400 degrees For 10-12 MInuTes

Lovely wITh a bIT of buTTer and chowder

The MaTh

Altogether, How many cups of Ingredients are does this recipe make?


1/4 cup waTer                                 1/4

1/2 Cup Mashed PoTaToes            1/2

1/4 Cup BuTTer (sIc. PaT               1/4

1/4 cup Brown Sugar                      1/4

1 Cup Milk (sIc. Almond Milk)          1

4 Cups WhITe Flour                         4


1 & 1/2 Tsp SaLT             

Answer:  6 and 1/4 cups of Ingredients; and 1 and 1/2 Tsp

Question:  How many Teaspoons of Ingredients are in a cup?  If we really want  to add the small still, we would have to calculate that from a chart, or physically fill up a cup of salt, one tsp at a time. 

Have a quick gander at some our Learning Math Games:

Teaching Fractions with Chocolate

Graphing Classroom Activity

Roll to Win Investigation – Graphing Classroom Activity


Graphing is an excellent way to display data visually. Students will come in contact with a variety of data and ways to display this data over time. It is important that students understand that there are three main types of graphs used to display information. The three types of graphs are line graphs, pie charts, and bar graphs.



Finding Area – Classroom Exercises

Area is All Around Us!  Simple Geometry Classroom Activities


One challenge that students often face is realizing that math DOES actually relate to the real world and that they will actually use the information they are learning at some point in their life. Area happens to be one of those topics that students struggle to understand the reasoning behind.  Before introducing the topic of area, you may want to ask your students some engaging questions such as: “If you wanted to put new tile down on the floor, how would you know how much to buy?” or “If you wanted to put wallpaper on the walls of the classroom, how much wallpaper would you buy?”. These questions will be sure to get them thinking about how to calculate these answers.

Multiplication Can Be Simple! – with a handout game!

math11Multiplication is an operation that requires you to add another number to itself a certain number of times as indicated in the multiplication equation.

When students first start learning the concept of multiplication, it is more simple as time goes on for kids to learn. Memorizing multiplication facts works for some students but not for all! Some students need to learn by using different models and representations. When students have a conceptual understanding of multiplication and realize that it is connected to the real world, they tend to perform better on assessments. If a child is only ever taught isolated facts or memorized facts, they risk the chance of not understanding the meaning behind the objects they are multiplying. Knowing a variety of ways to solve multiplication problems will allow a student to figure out which strategy works best for them.


Identifying Polygons

Polygons can be defined as two-dimensional, closed figures that are described by the number of sides, length of sides, and the kinds of angles. Many polygons have a respective name depending on their description. Some common polygons you will work with include: triangles, squares, rectangles, trapezoids, quadrilaterals, rhombus, pentagons, hexagons, parallelogram, and octagons.

When learning the names of polygons, students can easily be confused by terms that are used interchangeably at times. For example, a quadrilateral is considered to be a four-sided figure. So one might easily confuse this by calling all quadrilaterals squares or rectangles. However, by definition a rectangle is a special quadrilateral because it has opposite sides that are congruent or the same length and each angle is a right angle that measures 90 degrees. Another special quadrilateral is a square. A square has four sides that are all congruent or the same length as well as four angles that are all right angles measuring 90 degrees. This becomes easily confusing for a student when they are trying to identify polygons by name and descriptors.

Adding and Subtracting Polynomials

Rocket Llacey April 2015Ever launched a Rocket and wanted to determine how High it goes? Polynomials can come in handy when trying to model the flight path of a Rocket.  Did You Know that  when shooting a rocket straight up in the air, the rocket’s path can be modeled using the polynomial equation: y = -16t2 + vt + ho? Yes, it is True. Using this Equation you can easily Determine when the Rocket will hit the ground and even how far the Rocket will shoot into the sky. The Height the Rocket will reach is dependent on the initial velocity of the rocket and the initial height.  (Rocket Photo:

Notice that the Rocket Equation does not involve the weight of the Rocket. As a Rocket is launched the initial Velocity allows it to overcome Gravity. However, eventually, that initial force from the launch dissipates and Gravity takes hold. A Rocket reaches its Maximum Height shortly before Gravity forces it back toward the ground.

Interested in learning more about Math related to Rockets? Check out this link:

Learn More!  Let’s try an example, using the Polynomial Equation: d = -16t2 + vt + ho. If a Rocket is launched with an initial Velocity of 50 meters per second off of the ground, how high will the Rocket be after 3 seconds?  Solution:  So, the Rocket will be 6 Meters off the ground. The Rocket is likely on its way back down toward the ground.

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Math and The Chromatic Scale: Loving Music, Loving Math

Gold Harp wiki Nov 7Harmony
occurs in music when two pitches vibrate at frequencies in small integer ratios.  Long ago, Greek people realized the concept of harmony occurred when sounds and frequencies are in rational proportion. i.e., One Octave is equal to when the frequency is doubled, and a tripling of frequency brings the key One Octave higher, and is called a perfect fifth. Though not knowing this in relation to “frequency”, ancient Greeks knew this in relation to lengths of vibrating strings;  (Why 12 Notes to The Octave?)


1/1  unison              C

2/1  octave              C

3/2  perfect fifth       G

4/3  fourth              F

5/4  major third         E

8/5  minor 6th           Ab

6/5  minor 3rd           Eb

5/3  major 6th           A

9/8  major 2nd           D

16/9 minor 7th           Bb

15/8  major 7th          B

16/15 minor 2nd          C#

Piano Player Creating and LearningThe most common conception of the chromatic scale before the 13th century was the Pythagorean chromatic scale. Due to a different tuning technique, the twelve semitones in this scale have two slightly different sizes. (En. Wikipedia. org/wiki/Chromatic_scale)  Thus, the scale is not perfectly symmetric.  Pythagoras, 13thC Greek mathematician, was famous in geometry for the Pythagorean theorem (en.  Wikipedia. org / wiki/Pythagoras).  The theorem states that in a right-angled triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is, a^2 + b^2 = c^2.  Pythagoras experimented with a monochord, noticing that subdividing a vibrating string into rational proportions produced resonant sounds. When the frequency of the string is inversely proportional to its length, its other frequencies are simply whole number multiples of the fundamental.  (En. Wikipedia. org/wiki/Chromatic_scale)

The term chromatic derives from the Greek word chroma, meaning color, where the total chromatic / aggregate is the set of all twelve pitch classes; an array being a succession of aggregates.  Shí-èr-lǜ (Chinese: 十二律 (twelve-pitch scale) is a standardized gamut of twelve notes. The Chinese scale uses the same intervals as the Pythagorean scale, based on 2/3 ratios (2:3, 8:9, 16:27, 64:81, etc.). The gamut or its subsets were used for tuning and are preserved in bells and pipes.  In China, the first reference to “the standardization of bells and pitch,” dates back to around 600 BCE.  According to ancient scroll/script literature, Pythagoras taught that music was not intended for entertainment, though for calming the mind and bringing about order from chaos of life and the universe using spiritual instruments.  Music of the Spheres is one of the phrases used to describe Ancient Greek Pythagorean Music.  Here is a sample of what this music sounds like:  This clip is a short educational video on a Pythagorean Tone Generator: Pythagorean Tone Generator:

James Hopkins, a student and practitioner of Pythagorean Monochords visually shows us his handmade Monochord Stringed instruments:

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Golden Mean Ratio: Egyptian Sculpture

Math in Ancient Sculpture is one of the most interesting and intriguing applications of mathematics, both formula equation and geometric structure.  This article examines the Golden Mean Ratio and Fibonacci FibonacciChamomile wikip Oct 7 2013  Nefertiti Neues Museum Wikip Oct 7 2015Sequence found in the Bust of Nefertiti in Egypt in the early 1900’s. Nefertiti literally, means, “the beautiful one has come”, and is the 14th-century BC Great Royal Wife (chief co-regent) of the Egyptian Pharaoh Akhenaten of the Eighteenth dynasty of Egypt, 1352 BC to 1336 BC.  The iconic Bust of Nefertiti is in Neues Museum of Berlin; though originally in Egypt (En.   Wikipedia.   Org / wiki/Nefertiti_Bust   (230px-Nofretete_Neues_Museum 2 .jpg)  During excavation of Nefertit’s Bust, one of the items found noted Thutmose as being the court sculptor of Egyptian Pharaoh Akhenaten, Nefertiti’s Royal Husband.  In Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, Stakho speaks of the Berlusian philosopher Edward Soroko’s attempts to determine what mathematical ideas were used in the creation of Her Bust, “…Harmony was the perogative of the Divine order that dominated the universe, and geometry was the main tool of its expression. 

Nefertiti, playing role of Goddess, thus Her image personifying the Wisdom of the world, must have been formed with geometrical perfection and irreproachable harmony, beauty and clarity.  As a matter of fact, the main idea of ancient Egyptian aesthetic philosophy was to glorify the eternal, the measured, and the perfect in a constantly changing universe…In his analysis, he found a harmonious system of regular geometric figures such as triangles, squares, and rhombi.”  (Alekseĭ Petrovich Stakho; Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science; Scott Anthony Olsen. Book. English. 2009; pp 57-59)  Soroko believed that the Bust of Nefertiti was created based upon the Geometries of the Golden Mean Ratio and the Fibonacci Sequence.  A graphical image of Her Bust, as was drawn by Soroko is shown in this link:  Highly recommended to click on this link to have a finer and more detailed idea of the geometries that Thutmose is thought to have in mind when sculpting Her Bust.  One can appreciate, in particular, how the front top vertex of Her headdress is in perfect alignment with Her Heart and Breast Bone.

Golden Mean Squares Wiki Oct 7 2014Depicted here, the Golden Mean Ratio, and in the geometries thought to be applied by Thutmose, the Egyptian Court Sculptor, in the creation of Nefertiti’s Bust:

n is to m, as m is to n − m,

or, algebraically,


n            =         m

_____            ______

m                     n – m

A more clear example of the pattern he believed Her Bust was created upon, the Fibonacci Sequence, can more easily be seen in examining nature’s pattern in pine cones, and seashells.  This ratio is also the most efficient mathematical equation for trees absorbing the most amount of sustenance from the sun’s rays, hence growing in a spiral.  In the same way, today, we can apply solar panels onto rooftops mimicking this pattern to most efficiently absorb the sun’s rays.  For further discussion on Fibonacci Sequence, visit here:

See in the photo at the top, the chamomile spiral, as well as in this link on “fibonacci-sequence” showing the pattern.  Adding two consecutive numbers from the sequence to equal the next one following, the basic mathematical sequence looks like this:

0,         1,            1,            2,            3,            5,            8,            13,            21,            34,            55,            89,            144,            233,            377, etc., etc.

That is:



2+1 (the “number” before)=3

3+2 (the number before)=5








etc., etc., etc.

Whereas the fibonacci sequence, cannot technically be exacted in real time, (similarly, there is no real exact end found as of yet to pi’s “3.14159265359…..”), mathematicians drew swirling squares around the spiral in the attempt to exact the formula – basically giving the left side of the human brain a way of understanding the right-brained infinite spiral, being that the left side of the brain requires finiteness / exactness to feel satisfied, shall we say.  

What kind of intriguing application can your class find to prove that math is interesting and fun?

We wonder if the geometries of Nefertiti’s Bust were also the same geometries of Nefertiti’s real-life human head.  The skulls of Ancient Egyptians, as well as those of Ancient Greece and other cultures, have definitely been shown to have geometrically perfect structures.  How can that be?

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Torus Math – The Doughnut

Torus Aug 28 wikipedia from png
Welcome Back to Math Class!  We trust all had a wonder-filled summer filled with fun spherical geometric shapes and bursting color!  Our Back-to-school math topic this autumn is about the math magic of the Torus. A torus is: the area and volume of, basically, a doughnut shape, shown here: 
The author of harmonic resolution; The Portacle, describes that the torus’ special form has been used to describe and/or represent a number of things in our “real” actual material world, as well as, our “imaginary” potential one; and is defined by two parameters: the radius of the torus (that is, the radius of the torus’s defining circle, measured from the origin) and the radius of the tube (the perpendicular distance from the defining circle to the surface of the torus). These are R and r, respectively (R > r).   -In this link, there is a Great fluid-moving image of a spinning torus, about 4/5th down the page.)  Wikipedia has a diagram of the torus as being the product of 2 circles, one sideways inside the other: See image to the left here: A torus is the product of two circles, in this case the red circle is swept around the axis defining the pink circle. R is the radius of the pink circle, r is the radius of the red one; Image drawn August 28, 2014:

Some say that Earth is neither flat or round, but spherical, as in that of a doughnut toroid or even an apple, curving down and curving upward at the North and South poles.  One image we find most interesting is just under the blue torus image, and it is a mutable image of a ring torus turning itself into a horn torus, and this is one of the images that mathematicians today are calculating that Earth is most closer to as a realistic geometric shape, than was once thought.  Earth, in this respect is today referred to as the tube torus, a shape that occurs often in Nature.  If you click on the link, you can see just to the right under the blue torus, a better mental picture of our reference: As the distance to the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally degenerates into a sphere;  Torus_cycles.svg August 28 wikipedia from png
(There are several on-line sources of the manifolds, 3 of which quote the same information:  Joyce, Wikipedia and Harmonic, all denoting that manifolds are today, a major branch of geometric study, a curved space of some dimension. For example, the surface of a sphere, and the torus (the surface of a doughnut), are both 2-dimensional flat-surface – perceived manifolds. The manifold itself is the background for some mathematical object defined upon it, as a canvas is the background for an oil painting. This kind of geometry, although very abstract, is closer to the real world than you might think. Einstein’s theory of General Relativity describes the Universe – the whole of space and time – as a 4-dimensional manifold.  Wikipedia has wonderful moving images of the manifold turning into the torus and back into a manifold flat surface again, and is worth having a gander for a more picturesque idea of what these terms are in Math, under the subtitle “ Flat Torus“:

Dominic Joyce, Universty of Oxford, (www. maths.ox. iterates that today’s popular geometric math is in manifolds, no longer Euclid’s triangles and circles, and says that popularity went out with the Arc!  We think he is referring to Noah’s Arc, which many would probably disagree about Noah Arc’s popularity because of all the beautyful animals on the Arc!  But that is digressing….Joyce says that, Space itself is not flat, but curved. The curvature of space is responsible for gravity….and that, Everything in the universe – light, subatomic particles, pizzas, and even, yourself – is described in terms of a geometrical structure on the space-time 4-manifold.  Joyce says that, according to some physicists, the universe is now considered to be a 12 (or 13, 15, or 17, by some!) dimensional nested manifold.

In “superstring” physics, the torus is known as the “perfect” shape…but String Theory we can discuss on another day!

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