During the late 1990’s Fox studios approached a Las Vegas magician, Val Valentino, with a proposal for a new television series. The planned shows would expose the secrets behind popular magician’s tricks. Magicians follow a ‘code’, pledging never to reveal the secrets behind their magic, however Valentino agreed and soon became the ‘masked magician’.
At the conclusion to the series Valentino explained his motivation, “…revealing the secrets will encourage kids to try magic rather than discouraging them”. While the resurgence of interest in magic over recent years is more likely associated with the Harry Potter phenomenon than Valentino, the series was extremely popular and is often repeated.
Despite their exposure, magicians continue to amaze audiences with physical and optical illusions. A parallel group of magicians now entertain audiences creating illusions of computational wonderment, they’re called Mathemagicians. Just like their theatrical cousins, Mathemagicians have a number of techniques, or should that be techniques of number? Some of the Mathemagicians’ routines incorporate a mentalist’s persona, an entertainer whose performance appears to be based on psychic abilities. It is this type of illusion that engages students most immediately and intensely. Phenomenally fast mental computation is the other art of the Mathemagician. Art Benjamin is a wonderful Mathemagician exchanging sleight of hand for cerebral gymnastics in the name of entertainment equating him to a human calculator.
Treating students to examples of mathematical – magic can increase motivation and engagement. Only a handful of individuals have the ability to perform at the level of a Mathemagician; however most students enjoy learning some of these illusions so they may perform these tricks on their friends and family. Most of the routines involve basic number theory or a combination of numerical patterns that can be explained using basic algebra.
The secret of nines
As an educational consultant I spend many hours working in different schools. On one occasion I was working with a school helping introduce Computer Algebra Systems. (CAS) In the morning I worked with a class doing some mathematical magic, creating the illusion that I could read the student’s mind, the mentalist persona. After several examples the sceptics began to question the illusion and soon discovered a pattern. We discussed the pattern, formulated a conjecture and then proved the result algebraically. This magic trick is a lesson on algebraic representation, expanding and simplifying.
One of the teachers observing the lesson asked if I could do the same with her class later in the day. That afternoon I repeated the lesson as requested. Students enjoyed the first ‘illusion’ but we didn’t need to repeat it. One of the students in the class ‘exposed’ the illusion. The classroom teacher was disappointed; I proceeded to question the student. The student admitted that they hadn’t figured it out for themselves; they had been discussing the illusion at lunchtime with their friends. Initially their classroom teacher was annoyed, until I asked “how often do your students talk about mathematics at lunchtime?” Students are more likely to remember or develop a deeper understanding when they discuss their learning with others. “Thought is not merely expressed in words, but comes into existence through them.” [Vygotsky 1996]
Mathematics is one of the few subjects that require students to engage in pure thought. Student objection is often verbalised through the familiar challenge “where are we going to use this?” Teachers often struggle to find an immediate purpose or context. According to Piaget, students learn new concepts through assimilation and accommodation. Many junior level secondary students are still at the concrete learning stage, these abstract concepts prove challenging and therefore demand a higher degree of motivation. Motivation is provided through the mathematical illusions. Students ask “How do you do it?” Many of them want to learn so that they can perform the same illusions on friends and family. Students are more likely to learn when they have a purpose.
One of the most valuable questions asked by students during such lessons is: “Does it always work?” Number tricks generally involve too many possibilities, testing them all would be time consuming. The necessity and power of algebraic proof comes to light.
The illusion used with the students appears on many web sites. The same illusion used in the lesson (and demonstrated in the Mathemagicians workshop) can be created using a series of PowerPoint slides. For a copy of these slides email: firstname.lastname@example.org
Step 1: Select a two digit number. Example: 65
Step 2: Add the two digits together. 6 + 5 = 11
Step 3: Subtract this quantity from the original number. 65 -11 = 54
Step 4: Locate the result on a page containing a mixture of numbers and apparently random adjacent images.
Step 5: The Mathemagician correctly identifies the image the student has selected. Note that it is not the number the student selected, only the adjacent image.
The images adjacent to the numbers are not completely random; at least every image adjacent to a multiple of 9 is not random. Once students have identified the pattern, they develop a conjecture:
“I believe all the answers are multiples of 9.”
The objective is to prove their conjecture. The proof requires some relatively simple algebra and illustrates how algebra can be used for proof and generalisation. Too much of the algebra taught in schools relates to algorithmic manipulation, simplification and procedural recall through rote. Rote learning, by definition is learning without understanding.
Step 1: Select a two digit number. 10a + b (Where a and b are single digit values)
Step 2: Add the two digits together. a + b
Step 3: Subtract this quantity from the original number. 10a + b – (a + b) = 9a
The result appears straight forward; however it is quite likely many students will not fully understand the significance without testing the result numerically. 9 x 1 = 9, 9 x 2 = 18 …
The inspiration for this problem comes from the American Mathematics Teachers Journal. The original problem is done by hand; the electronic and wireless medium of TI-Navigator makes the problem significantly more impressive. Students are asked to generate 8 random numbers. It is suggested to keep the numbers relatively simple as some mental computations will be required later, “perhaps restrict your numbers to two digit numbers”.
Students enter the random numbers on the calculator and then type the word “Magic”. The result is a matrix (which has been predefined) that appears to contain a selection of random numbers with little or no apparent relationship with the original random numbers. A sample of the magic square created is shown below:
Ask students to add the numbers in the magic square; “don’t use the calculator or let the calculator know which numbers you selected”. The teacher then tells the students: “I calculated everyone’s answers and sent it to your calculator. Write the word answer on your calculator.” Students are amazed that whilst they have different answers to each other, the calculator knows the answer they computed. It is left to the reader to figure out the secret behind this trick! The trick is suitable for students in junior secondary.
Developing Human Calculators – Perfect Squares and Difference of Perfect Squares
There are many ways to calculate the product of two numbers and even more for squaring numbers. Students are generally taught using traditional methods; however there are many other ways this can be done both visually and mentally.
The diagram shown opposite illustrates how drawing lines to represent the tens and units values can result in a very simple approach to multiplication. Vedic mathematics techniques can also be employed (Workshop demonstration). These techniques can also be validated using simple algebra.
Sample Vedic Mathematics result: (Reference number = 100)
98 x 96 100 – 98 = 2 & 100 – 96 = 4
2 x 4 = 8
94 x 100 = 9400 96 – 2 = 98 – 4 = 94
98 x 96 = 9408. 9400 + 8 = 9408
An algebraic proof of the above approach is left to the reader. The use of CAS means that students can focus on producing a ‘general solution’ rather than algebraic manipulation.
Difference of perfect squares can be applied to squaring numbers such as 15, 25, 35 … A wonderfully simple paper folding activity can help students remember how ‘difference of perfect squares’ works, then supported with the algebraic approach and applications to number.
(a + b) (a – b) = a2 – b2 Let a = 35 and b = 5
(35 + 5)(35 – 5) = 352 – 52 Substitute
40 x 30 = 352 – 25 Simple calculation 40 x 30 = 1200
1200 + 25 = 352.
Perfect Squares can also help, particularly for numbers such as 31, 41, 51 …
(a + b)(a + b) = a2 + 2ab + b2 Let a = 30 and b = 1
(30+1)(30+1) = 302 + 2x30x1 Substitute
312 = 900 + 60 + 1 = 961
With a little practice it is possible to square numbers such as 126.
(125 + 1)(125 + 1) = 1252 + 2 x 125 x 1 + 12
[Note: 1252 = (125 – 5)(125 + 5) + 25 = 120 x 130 + 25]
1262 = 15625 + 250 + 1 = 15,876
It may seem complicated to apply different approaches to multiplication problems when we have a ‘one algorithm suits all’ solution already. However, students already have valid techniques to calculate problems such as: 100 – 98 and for slightly more advanced students: 112 – 98. Think about the way that you calculate the answer. Asking students to describe their thinking, metacognition, when solving problems such as this is a very powerful learning tool in itself.
There are many more magic tricks that involve mathematics. There are card tricks which revolve around numbers with different bases and others that rely on divisibility tricks. Try a couple with your students and challenge them to learn the tricks to try on their friends and family. Students remember the mathematics as it is associated with the thing they want to remember, the magic trick. Of course it is all just a trick to get students to engage in mathematics!