In this Newsletter:

1. The equal sign: more trouble than it’s worth?
2. Application: Ellipses and composite trigonometric graphs
3. Back to school tips
4. Friday math movie: Right brain math
5. Final thought: Just do it!

1. The equal sign: more trouble than it’s worth?

Suitable for: Everyone, since we all come across equal signs every day.

2. Equation of Time, an application of ellipses and composite trigonometric graphs

Suitable for: Everyone. You will come across conic sections and trigonometric graphs at some point in your math career.

3. Back to school tips

Many of your are starting school again very soon. Here are some earlier articles you may find useful:

• Ten Ways to Survive the Math Blues
• How to understand math formulas
• How to learn math formulas

4. Friday math movie: Right brain math

Suitable for: Everyone.

Never could master those times tables? This right brain approach might be a solution for you. Watch the video: Right Brain Math – Times Table

5. Final thought: Just do it!

As the school year begins (for many of you), it’s a good opportunity to consider how you are going to make the most of it.

The research is in: multi-tasking just means we do a lot more things, but none of them well. (See Is multi-tasking a myth? from the BBC).

Want to do well at anything (not only math)? There’s a simple solution – focus.

Until next time.

Related posts:

1. IntMath Newsletter – resources, modeling and mistakes
2. Intmath Newsletter – resources, summation, Riemann, census
3. The IntMath Newsletter – 1 Nov 2008

First, let’s investigate the properties of an orbiting body.

In the following graph, the green ball represents the Earth rotating around the (red) Sun, which is at a focus of the ellipse.

This is greatly exaggerated so you can see what is going on.

Consider how fast the sun would appear to be moving when we are close to it, compared to when we are way out near the second focus of the ellipse (labeled F₂.)

[In fact, the Earth's orbit around the sun is very close to a circle, but it is still elliptical. See more on the Ellipse.]

The Equation of Time

Many of us don’t take much notice of the motion of the sun, moon or planets. (Let’s face it – most of us live in places that have bad air and light pollution so we can hardly see the sky).

Sundial [Image source]

Consequently, we assume the Earth’s motion around the sun is uniform and that it takes exactly 24 hours for the sun to come back to the same spot above us each day.

But this is not so. There is as much as 30 minutes variation between the position of the sun relative to the stars at the same time of day, throughout the year. That is, the sun can be "behind" by as much as 14 min 6 sec (around 12 February each year) and up to 16 min 33 sec "ahead" (around 3rd Nov each year).

There are 2 reasons why this is so:

1. The Earth revolves around the Sun along an ellipse, not a circle. As we saw above, this means the Earth does not travel around the sun at a constant speed.
2. The Earth is tilted at 23.44 degrees

If the Earth rotated around the sun in a perfect circle and there was no tilt on the Earth’s axis, we would see the sun overhead in exactly the same position every day.

The ancient Greeks knew about this issue but because they didn’t have accurate clocks, they were not too concerned. By the 17th century, with the invention of pendulum clocks, knowing where the sun should be at any time of the year became critical for accuracy in navigation at sea.

The Equation of Time

The Equation of Time takes the above 2 factors into consideration and can tell us how far ahead or behind the sun will be relative to the stars.

An explicit function that is a good approximation to the Equation of Time is as follows, where d is the day of the year and the resulting value is in terms of minutes of variation):

Time variation = -7.655 sin d + 9.873 sin(2d + 3.588)

This is made up of 2 components:

Variation due to elliptical orbit = 9.873 sin(2d + 3.588)

This contributes 9.873 minutes to the variation in the sun’s position, and has period half of one year:

The second component:

Variation due to Earth’s tilt = -7.655 sin d

This contributes 7.655 minutes to the Equation of Time, and the period is 1 year (365.25 days).

[See more on Period of Trigonometric Graphs.]

When the 2 components are added together, we obtain:

When we add ordinates of trigonometric functions like this, we obtain what is called a composite trigonometric curve. (See more on Composite Trigonometric Curves.)

(You can see how the Equation of Time is derived at Equation of Time.)

Conclusion

The Equation of Time is a neat real-life application of conic sections (the Ellipse) and Composite Trigonometric Curves.

Related posts:

1. Hubble math
2. MS Equation Writer – not very impressed
3. H1N1 and the Logistic Equation

A common problem

Start with 7 marbles, add 3, add another 3, then subtract 8. Many students will write this for their answer:

7 + 3 = 10 + 3 = 13 − 8 = 5

Now, each of the operations is done correctly, but the whole thing is actually not correct since the first expressions (7 + 3 and 10 + 3) don’t equal the last expression (13 − 8) or the final number.

This is called a "running equals" and it is quite wrong.

In the article Students’ understanding of equal sign, we learn that 70% of middle grade students in the US had misconceptions about the equals sign (Research by Robert M. and Mary Capraro of Texas A&M University.)

As an example, they cite the following question:

4 + 3 + 2 = ( ) + 2

Many students will incorrectly add everything on the left, put that in brackets, then proceed to add the final 2.

4 + 3 + 2 = (9) + 2 = 11

The question really requires the following:

4 + 3 + 2 = (7) + 2 = 9

Why do students make such mistakes?

Background

One of the first things we learn in math is the equal sign. We count 4 blocks and add 2 blocks and get 6 blocks. Some (well-meaning) person tells us we can write this as follows:

4 + 2 = 6

Did you realize that the equals sign is a relativelly recent invention, and that mathematicians did just fine without it for thousands of years?

The equals sign, "=" was invented by a man from Wales, Robert Recorde in the mid-16th century. He used equal-length parallel lines because "nothing could be more equal".

Before the invention of the "=" symbol, mathematicians employed useful and expressive words to indicate what was going on. So when adding, they used "gives" or "yields". Other early symbols for equals included “æ” (from the Latin “aequalis“) and 2 vertical parallel lines, ||.

Over the years, several different meanings have evolved for the "=" sign, and this can increase student confusion.

Meaning #1 – result of performing an operation

In the 4 + 2 = 6 example, "get 6 blocks" in my English sentence is written as "= 6" when translated into a statement using symbols only.

Meaning #2 – decomposition

What if we flip the above statement and write it as follows?

6 = 4 + 2

Now the "=" sign doesn’t represent the same concept as our "=" sign in the earlier 4 + 2 = 6 example.

Now it means we start with 6 and we have decomposed it into 4 + 2. But we could also decompose 6 in any of the following ways:

6 = 3 + 3
6 = 2 + 4
6 = 5 + 1
6 = 0 + 6

This time we have an infinite range of answers. Clearly "=" here is used differently than in the 4 + 2 = 6 example, where we had one possible answer (assuming we are only talking about real numbers, of course), while the second set of examples starts with a single number and suggests a very open-ended set of solutions.

Meaning #3 – equivalent value of 2 operations

Now, let’s look at another case.

5 + 9 = 7 × 2

Here the "=" sign means something different again. Now we need to do something to the expression on the left (add them) and do something different with different numbers on the right (multiply them) and after all that, we can conclude that indeed both sides have the same value.

Meaning #4 – equivalence in sets

In set theory, "equal" is not the same as "equivalent".

For example, the set {1,2,3} is equivalent to the set {a,b,c} since they both have the same number of elements (the same cardinality), but they are not equal since they don’t contain exactly the same elements.

Also, we can write {1, 2, 3} = {2, 1, 3} (they are equal) since the elements in the set are exactly the same, but the order doesn’t matter.

Meaning #5 – equivalence in equations

We can have equivalent equations. For example, the following 2 equations are equivalent since they have the same solution set:

x + 2 = 7
3x = 15

Both equations give us x = 5. We can’t say we have "equal equations", but we do use the term "equivalent".

Meaning #6 – equivalence in ratios

Two fractions can be equivalent:

They look different but they have the same value.

Notice I used the "3 horizontal lines" symbol (≡) for "equivalent". We can also say "equal" in this example.

Meaning #7 – as a limit

This next one causes a lot of debate. Start with 0.9. Then, take the larger number 0.99. We have moved closer to 1, correct?

Now, keep going. 0.99999… Clearly, if we go (infinitely) far enough, we will end up with this result:

0.999… = 1

This employs a very different concept to the first 3 meanings of "=" given above. Is it exactly equal to 1? Don’t we need a "limit" statement of some kind?

Here’s another one. Being irrational, pi cannot exactly "equal" some number. Or can it?

We can show that pi has the following value:

Should we write "=" or "≈" (approximately equal)?

Meaning #8 – use in proofs

Here’s another place where many students come unstuck – proving trigonometry identities. Say the question is:

Prove

Many students attempt a solution something like this:

Now essentially the steps are correct, but it is difficult to follow the logic. Should we go left-to-right, or top to bottom, or a mix of the two?

What the student has done is to work down the left side and right side in a parallel fashion until he got something that looked the same on both sides.

It’s not the best proof, since there is no clear statement of what we start with and what we are doing in each step.

Some possible solutions for the "=" sign problem

Computer languages and computer algebra systems use different symbols for the different meanings of the equals sign.

This might be a good solution for mathematics.

In PHP computer language, for example (which IntMath and squareCircleZ uses), "=" is used to assign the value of a variable, while "==" is used to check for equality.

So for example, the following statement means variable "a" will have value 5 from this point on in the program.

a = 5

If we need to test to see if some condition is true, for example whether variable "b" has value 3, we would write:

if ( b == 3 ) {
[do something...]
}

In the computer algebra system Mathematica, the "==" sign is used for solving equations, as follows:

Input: Solve[2x == 6, x] Output: {x->3}

Note the "->" symbol for equality in the output.

For comparison, the rival software Mathcad has 4 different equal signs:

Assignment (:=) To assign a value to a variable, you need to use the "colon equal" notation which looks like this:

a := 5

Display a value (=) To display the value of variable "b" in Mathcad, you just type "b=" and the value will appear:

b = 8

Symbolic equality (=) To evaluate a symbolic expression you use the "bold equals" symbol:

b + b = 2b

Global assignment (≡) This is the "identically equal to" symbol from mathematics. This is used when you want to fix a variable throughout a whole math worksheet, not letting it be changed by other variables:

c ≡ 8

This is a different sense from that used in mathematics, however. But as the identity symbol suggests a "strong equals", perhaps it is appropriate.

Part of the problem?

This is a screen shot from a video that extols the virtues of interactive white boards for math teaching.

The instructor’s question is "Evaluate the equation". Can you really "evaluate an equation"?

You can solve an equation for some variable, or you can state that expressions are equal in an equation, but you can’t evaluate an equation.

Source: eSchoolNews

We need to be careful how we express mathematics questions since this is where misconceptions can arise.

Summary

So why do students make mistakes with the equal sign? It’s most probably because there are several nuances in meaning for the equal sign.

Perhaps we need to consider using a wider range of symbols for the different types of equals. One place to look is the notation used in computer languages or computer algebra systems.

Related posts:

1. Equal areas of a circle gives nice art
2. IntMath Newsletter: Time, equals and resources
3. Katrina’s message to Bush: sign Kyoto

However, there are many students who have a more intuitive and holistic view of the world (they are more “right brained”), and have trouble understanding the math teacher’s explanations (and algorithms).

Are you right brained?

This short video gives you an indication about whether you are left- or right-brain dominant.

If she only goes in one direction for you, try looking at the video by first covering your left eye so you perceive it using your left hemisphere (you’re looking at it with your right eye), then cover your right eye, so your right hemisphere is busier.

Right brain times table

Tom Biesanz presents an interesting visual method for understanding the patterns in the times tables. He claims it appeals to right-brain dominant students.

Of course, none of us is completely right- or left-brained, but like many things, we certainly have a preference.

Related posts:

1. Friday Math Movie – Deep Brain Stimulation
2. Friday math movie – Pump up your brain
3. Friday math movie – Daniel Tammet – The Boy With The Incredible Brain

In this Newsletter:

1. Number Sense
2. Math resource – Wolfram|Alpha
3. Math tip: Irrational numbers
4. Taxicab Geometry
5. Friday math movies: (a) Smarter Math (b) Murderous Math Tricks
6. Final thought – Do something meaningful

1. Number Sense

Suitable for: Everyone.

2. Math resource: Wolfram|Alpha

Wolfram|Alpha is a "computational knowledge engine". It’s an excellent resource for math students (and instructors too!).

Did you know it can answer a wide range of questions about math as well as do actual algebra questions?

For example, it will solve equations for you (When you get there, click where it says "Show steps"):

solve x^2 – 2x – 3 = 0

It will perform integrals written in plain English:

integral from 0 to 4 of e^x sin(x)dx

Here are 10 more things from their blog (and these activities are not only for "kids"):

10 Fun Questions Kids Can Answer with Wolfram|Alpha

3. Math tip: Irrational Numbers

Suitable for: Everyone.

3. Taxicab Geometry

Suitable for: Everyone. This starts with a simple concept, and leads on to some interesting considerations for calculus.

4. Friday math movies

(a) Smarter Math: Equations for a smarter planet

Suitable for: Everyone.

Here’s a positive message about math from IBM. We also learn about their “The Internet of Things”. There are too many neagative messages about math in the media! Watch the video: Friday math movie: Smarter Math: Equations for a smarter planet

(b) Murderous Math Tricks

Suitable for: Everyone.

Here’s some fun. How do these math tricks work?   Watch the video: Friday math movie: Murderous math tricks

5. Final thought – Do something meaningful

Steve Jobs, CEO of Apple and very successful developer of iPods, iPhones and iPads, once said that his goal was not to die the richest man in the cemetery. It was to go to bed at night thinking that he and his team had done something wonderful.

Those who do meaningful things with their lives tend to be the happiest. This almost always involves helping others.

Until next time, enjoy whatever you learn.

Related posts:

1. IntMath Newsletter – WolframAlpha, point of inflection
2. Taxicab Geometry
3. IntMath Newsletter: Time, equals and resources

Humans (as well as many animal species) are born with a "number sense", according to Stanislas Dehaene in his book The Number Sense – How The Mind Creates Mathematics.

This means we already have a "feel" for number quantities long before our parents teach us, or before we ever go to school. And animals also have a limited number sense, but not quite as sophisticated as that demonstrated by Maggie in the above video. (Actually, Maggie can’t really count. What’s going on there is what is known as the "Hans Effect". See Clever Hans.)

As is often the case, the reality is more remarkable than the fiction. Animals can certainly count, and just like us, they are much better with smaller numbers (1, 2, 3) but become more vague and prone to error as the numbers get larger (4 and onwards).

Counting vs Subitizing

Look at these 2 pictures. How many circles are there in each?

Now think about how you counted the circles.

For the left one, you most likely just looked and immediately concluded "3", but for the one on the right, you most likely had to count them one-by-one to make sure you got the correct number, right?

In the case where we know the number of objects without individually counting them, we are said to be "subitizing", a term coined in the late 1940s by E. L. Kaufman and others.

We share this subitizing ability with many animals.

Animal Math

A rat can be trained to pull a lever a set number of times to get a food reward. It "knows" there is a difference between 2 pulls and 3 pulls, and soon gets plenty of reward for the correct number of pulls. But if the required number of pulls increases, the rat becomes increasingly innacurate and tends to pull the lever random times in the hope of being correct.

Similarly, chimpanzees can tell the difference between the number of objects in different piles. An interesting observation is that the chimp can easily tell that 6 objects is "more" than 2 objects, but commits more errors when trying to compare 5 objects with 6 objects. This is called "distance effect" and the task becomes even more difficult when the number of objects increases (for example comparing 9 and 10 is much more difficult for the chimp than comparing 1 and 2 objects, even though the difference between them is the same). This is called the "magnitude effect".

It is the same for us. We can easily tell the difference between small groups of objects (or a small group compared to a large group), but we have difficulty comparing 2 large groups, since we can only be sure of our conclusion if we count them one-by-one.

Babies, too

What is interesting is that babies who are just a few days old exhibit subitizing. That is, they can distinguish between different numbers of objects and sounds. The way psychologists test this is to sit the babies on their mothers’ laps and repeat sounds over and over. If the sounds consist entirely of 2-syllable words, the baby becomes bored quite quickly, but if a 3-syllable word is inserted, the baby immediately becomes attentive again. This demonstrates that babies can "count", at least in a rudimentary way.

1, 2, 3, many…

Let’s go back to the simple task of counting dots, like in my 2 pictures earlier.

Since the 1880s, psychologists have known that counting 1, 2 or 3 objects was very quick and very accurate (subitizing), but for counting larger groups of objects, the amount of time required and the number of errors increases dramatically.

In the following chart, we see the result of getting people to count dots as quickly as they can. For 1, 2 and 3 dots, the time required is around 0.6 of a second, whereas there is a linear increase from there to 6 dots. There is a similar increase in the error rate.

One explanation for the above results is that we perceive 1, 2 or 3 objects using a "parallel processor", but we need to use a different serial mechanism to keep track of higher numbers of objects.

Number and Language

Consider for a moment how languages deal with 1, 2 and 3 differently from the rest of the numbers.

In English, for example, we use "th" and say fourth, fifth, sixth and so on for all the ordinal numbers therafter. But we have different endings for the first 3 ordinals: first, second, and third.

In some tribal languages, the only numbers used are "one", "two", "three" and "many".

So what is "Number Sense"?

Number sense involves being aware and having a reasonable understanding of the following concepts:

• The differences between numbers and their symbols
• Numbers as used to describe:
  • How many (cardinal e.g. "5 cats");
  • The order of things or events (ordinal, e.g. "the 5th book from the left"); and
  • The name or identity of something (nominal e.g. "his phone number is 9226-7841")
• The relationship between numbers (big-small, before-after). The concept of number line is important here.
• The effects of operating on numbers (adding, subtracting, etc)
• Estimation and rounding
• Being able to use relationships between numbers to simplify calculations (e.g. 79 + 45 = 80 + 45 − 1 = 125 − 1 = 124)

There are some really interesting cases where people have suffered from some brain trauma and have lost their “number sense”, or perhaps their ability to count. They will be able to get up to 3 just fine, but lose track very quickly after that.

Further Reading

I recommend:

• The Number Sense – How The Mind Creates Mathematics by Stanislas Dehaene. (You can see some of it via Google Books)
• Teaching Mathematical Concepts
  (This is actually a resource for teaching blind children, but there is some good advice in there.)

Related posts:

1. Making Sense of Online Learning
2. Yahoo and Google – who has the youngest audience?
3. How to draw y^2 = x – 2?

Let’s start with an interesting question.

Can you construct the length √7 just by using a measuring stick, a pencil and a set-square? (A set-square is a triangular device used for producing right angles.)

We’ll answer this question a little later. Some background first.

Length of the Hypotenuse

In the 5th century B.C., mathematicians were fascinated – yet exasperated – with irrational numbers. They believed the only meaningful numbers were the natural numbers (1,2,3,…) and any ratios involving these numbers (like 5/2, 7/9, etc). So they only accepted rational numbers and any other numbers were "unmeasureable".

Pythagoras (or someone in his metaphysical school of mathematicians) had shown the famous result that for a right angled triangle, the area of the square on the hypotenuse (in green below) equals the sum of the areas of the squares on the other 2 sides (the 2 light red squares).

You normally see Pythagoras’ Theorem written as follows, where c is the hypotenuse and a and b are the lengths of the other 2 sides:

a² + b² = c²

In my diagram above, we have the common 3-4-5 triangle. Each number is an integer and for the ancient Greek mathematicians, this presented no problem.

However, since they believed irrational numbers did not exist there was a problem when they extended the Pythagorean formula to other values.

The length of the hypotenuse involved a square root:

Depending on the values of a and b, we could easily get irrational values for c. How could they measure these distances if they didn’t actually exist?

Theodorus of Cyrene

Theodorus of Cyrene was a 5th century B.C. mathematician and was born around 100 years after Pythagoras. (Cyrene is now called Shahhat, in Libya.)

He apparently proved that the square roots of 2, 3, 5, 6 and so on up to 17 were all irrational, except the perfect squares 4, 9, 16. (Unfortunately we no longer have the proofs.) He also went on to construct these supposedly non-existent distances.

He proceeded as follows.

Start with a right triangle with equal sides 1, giving a hypotenuse of √2 (which of course was a problem, because this distance didn’t officially exist):

Then, extend a line with length 1 unit (using your 1-unit measuring stick) at right angles to the first hypotenuse as follows. This gives us the length √3 after we apply Pythagoras’ Theorem to the new triangle.

Do it again, and you now get the length √4 = 2. Theodorus had discovered one hypotenuse with a rational number length.

He kept going and found that the next one to have a "rational" length was √9 = 3.

He continued on to √16 = 4, constructed one more, √17, then stopped.

And so now you know how to construct the square root of any number using a straight edge, a pencil and a set-square.

Eudoxus

It was one hundred years later when the Greek astronomer Eudoxus (around 370 B.C.) concluded that because we can measure irrational distances (as we did above), then irrational numbers must exist. Problem solved.

Conclusion

It’s interesting that throughout history, people have yelled "Impossible!" when some new type of number was proposed. But subsequently, mathematicians have shown that not only are many of those numbers possible, but they have proved to be very useful.

Apart from irrational numbers as we discussed above, people originally did not believe in the existence of the number zero, imaginary numbers and "infinitesimals" in calculus. But in each case, they have been accepted as true numbers and used in many real applications.

Related posts:

1. Pythagoras
2. Your body in numbers
3. Today is “e” day

A taxi driver needs to go from point C to B. She decides to take the following route (depending on her local knowledge of traffic and one way streets and so on), where she heads north along CD, turns East for DF, then North again, and so on until she reaches B.

Or she could have more simply travelled from C to A, then A to B. If CA = AB = 1 km, then the total journey is 2 km, no matter which route she took. (Distance CD = 200 m and there are 10 such blocks, so 2 km is the total distance.)

This gives rise to an interesting type of geometry called Taxicab Geometry, first proposed by Hermann Minkowski in the 19th century.

How Many Routes?

There are clearly many different ways of going from C to B. If we assume she is an honest taxi driver (and doesn’t go away from B at any time), then she can only travel North or East.

Let’s consider her first decision. She can only go North or East, so 2 choices:

Now, looking at the first 4 city blocks, we see there are 6 choices:

Continuing, we find the following number of possible routes as we add more rows and columns of city blocks:

2 (for 1×1), 6 (for 2×2), 20 (for 3×3), 70 (for 4×4), …

You may recognise these as being the middle number of each second row in Pascal’s Triangle.

This is also connected with counting theory and we can show that for our 10×10 grid, our taxi driver will have

²⁰C₁₀ = 184,756 choices for getting from C to B.

An interesting question arises

Let’s now take a route that gives us a more diagonal shape, as follows.

Once again, our total distance is going to be 2 km.

But for interest, let’s imagine we can actually travel along lanes that pass in the middle of the city blocks as follows.

Our total distance is still 2 km.

But now let’s push it even more, and take many tiny zig-zags as follows. Our total distance is still 2 km.

But what is the limit of this process? Won’t we actually be smoothly travelling up the diagonal, since we can’t even steer the car North then East, since the steps are too small?

And now we have the interesting question. What will be the distance the taxi travels up this smooth diagonal?

Well, √2, of course, from Pythagoras’ Theorem. So the distance from C to B is no longer 2 km, but around 1.414 km, or 0.586 km less than 2 km!

Where did that 0.586 km go?

Related posts:

1. Solar Taxi vs Oil Storm
2. Safety last
3. IntMath Newsletter – Earth geometry, Board math and Obama math

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1. Friday Math Movie – Google Tricks, Hacks and Easter Eggs
2. Friday math movie – Pi Day
3. Friday math movie – Martin Gardner

According to IBM’s “Smarter Planet”, the 3 ways they want to contribute to a smarter planet are:

1. Instrument the world’s systems
2. Interconnect them
3. Make them intelligent

The following video explains how math can make our planet more intelligent. (I was worried he was going to push Internet fridges, but I was spared.)

One has to wonder – is urbanization such a great idea…?

Related posts:

1. Friday math movie – Pump up your brain
2. Friday math movie – Martin Gardner
3. Friday math movie – 2009 Halloween Math Class