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The pyramid collection.



In this article I would like to present readers with some very useful activities, ideas and problems. I have used these at various times over the years and I think that, together, they make a very interesting collection. These pieces have come together from different sources, but like a jigsaw A Web server from the W3C that incorporates advanced features and uses a modular design similar to the Apache Web server. Jigsaw supports HTTP 1.1 and provided an experimental platform for HTTP-NG. See HTTP-NG and Amaya. , they complete the picture of my understanding of the numbers inside the building of triangles and pyramids.

In NSW NSW New South Wales

Noun 1. NSW - the agency that provides units to conduct unconventional and counter-guerilla warfare
Naval Special Warfare
, with the new K-10 Syllabus A headnote; a short note preceding the text of a reported case that briefly summarizes the rulings of the court on the points decided in the case.

The syllabus appears before the text of the opinion.
 now being implemented, most of this material would be best placed in the Number strand. The activities on the pyramids, cairns Cairns, city (1991 pop. 64,463), Queensland, NE Australia, on Trinity Bay. It is a principal sugar port of Australia; lumber and other agricultural products are also exported. The city's proximity to the Great Barrier Reef has made it a tourist center. , apex card trick and Pascal's triangle Pas·cal's triangle  
n.
A triangle of numbers in which a row represents the coefficients of the binomial series. The triangle is bordered by ones on the right and left sides, and each interior entry is the sum of the two entries above.
 all come under the large umbrella of what is called Additional Content in the new syllabus. This is certainly 'material in order to broaden and deepen deep·en  
tr. & intr.v. deep·ened, deep·en·ing, deep·ens
To make or become deep or deeper.


deepen
Verb

to make or become deeper or more intense

Verb 1.
 students' knowledge, skills and understanding, to meet students' interests or to stimulate student interest in other areas of mathematics' (Board of Studies, NSW, 2002, p. 14).

Senior students, doing their Preliminary and HSC HSC - High Speed Connect  Courses in General Mathematics (non-calculus course) (Board of Studies, NSW, 1999) could also benefit from the use of the material. The activities on ziggurats and pyramidal numbers (Math.) certain series of figurate numbers expressing the number of balls or points that may be arranged in the form of pyramids. Thus 1, 4, 10, 20, 35, etc., are triangular pyramidal numbers; and 1, 5, 14, 30, 55, etc., are square pyramidal numbers.

See also: Pyramidal
 also fit in nicely with this syllabus in the components of Measurement: M2 (applications of area and volume) and Algebraic 1. (language) ALGEBRAIC - An early system on MIT's Whirlwind.

[CACM 2(5):16 (May 1959)].
2. (theory) algebraic - In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements.
 Modelling: AM4 (modelling linear and non-linear relationships).

Students preparing for the Mathematics (Calculus calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.  course) and Extension One (Advanced Calculus course) will benefit from investigating sigma notation notation: see arithmetic and musical notation.


How a system of numbers, phrases, words or quantities is written or expressed. Positional notation is the location and value of digits in a numbering system, such as the decimal or binary system.
 for each of the layers in the solids in ziggurats as part of the Sequences and Series topic. The formulae also provide some visual examples of 'Mathematical Induction' in the Extension One course.

Activity one: Problem solving problem solving

Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error.
 pyramid

There are many pyramid puzzles A pyramid puzzle is a mechanical puzzle (assembly puzzle) consisting of two or more component pieces which fit together to create a pyramid. History
Unknown origin. Children of American pioneers played with simple toys such as these.
. The two shown here are adapted from an idea in the British magazine, Figure It Out (Nexus Media Ltd, 17 September 1998, pp. 9, 30, 60). This magazine and other logic puzzle A logic puzzle is a puzzle deriving from the mathematics field of deduction.

This branch was produced by Charles Lutwidge Dodgson, who is better known under his pseudonym Lewis Carroll, the author of Alice's Adventures in Wonderland.
 publications are a great source of stimulating materials for you and your students.

I have made up two examples similar to the actual ones in the magazine and your task is to fill in the missing numbers in the grid In the Grid is a game show that airs on UK broadcaster Five at 6.30pm week nights. It first aired on Monday 30 October 2006.

In the Grid is hosted by Les Dennis and is produced by Initial West, one of the Endemol UK companies.
 using the numbers already in the grid. The number in each brick is the 'added total' of both the bricks that it is sitting on.

Pyramid one

[ILLUSTRATION OMITTED]

Give the pyramid a try. As a teacher you often need to put yourself in the same position as your students in order to be able to help them and interest them in problem solving. I hope that the number of different methods that can be used will surprise you. Do not forget to discuss all the strategies that were used by you and your students on this problem. You can quickly make up similar questions and use the same idea again.

Discussion

a) Did you 'guess and check' with numbers in the bottom row with the two smaller triangles and the numbers 23 and 20 at their apex?

[ILLUSTRATION OMITTED]

b) Did you use algebra algebra, branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as  in the same triangles? Say letting the number in the bottom row, second from the left be x and then use (x + 9) + (x + 2) = 23?

c) Did you analyse an·a·lyse  
v. Chiefly British
Variant of analyze.


analyse or US -lyze
Verb

[-lysing, -lysed] or -lyzing,
 the whole triangle giving each number in the bottom row a letter? For example, a, b, c, d, e, f. The apex number at the top of the pyramid is a + 5b + 10c + 10d + 5e + f.

d) Did you need to fill in the whole pyramid just to find 152 in the apex brick of the largest triangle?

e) What different tactics did you need to use to find the numbers in the rest of the bricks?

Solution

[ILLUSTRATION OMITTED]

Here is another pyramid to solve.

Pyramid two

[ILLUSTRATION OMITTED]

Solution

[ILLUSTRATION OMITTED]

Activity two: Cairns

A similar activity is to work from the top down. A few years ago these Complete the Cairn puzzles were in the Brain Games section of The Australian Magazine, inside The Australian newspaper.
   Each of the stones in the upper rows of the
   cairn sits upon two lower stones. The
   number in each of the upper stones represents
   the difference between the numbers in
   the two stones on which it rests. What are
   the five two digit numbers in the bottom row
   of stones? (Each of the digits 0-9 is used,
   once only, in this row.)
   (Moodie-Bloom, 1995, p.70)


The picture of the cairn shows how similar puzzles can be drawn up quickly on cardboard, overhead, blackboard (1) See Blackboard Learning System.

(2) The traditional classroom presentation board that is written on with chalk and erased with a felt pad. Although originally black, "white" boards and colored chalks are also used.
 or whiteboard The electronic equivalent of chalk and blackboard, but between remote users. Whiteboard systems allow network participants to simultaneously view one or more users drawing on an on-screen blackboard or running an application. .

Cairn one

[ILLUSTRATION OMITTED]

See what tactics you and your students use to complete the cairn.

Cairn two

[ILLUSTRATION OMITTED]

Solutions

Cairn one

[ILLUSTRATION OMITTED]

Cairn two

[ILLUSTRATION OMITTED]

Hope you do the second one more quickly than I did--I kept making some wrong decisions!

Activity three

a) The apex card trick

This is another classic puzzle that unlocks the key to Pascal's Triangle. Martin Gardner Martin Gardner (b. October 21, 1914, Tulsa, Oklahoma) is a popular American mathematics and science writer specializing in recreational mathematics, but with interests encompassing magic (conjuring), pseudoscience, literature (especially Lewis Carroll), philosophy, and religion.  has been an absolute inspiration to me over the years. If you have never read or used any of his books please start by chasing up one of his many books. From Chapter 15 in his book, Mathematical Carnival (Gardner, 1975) comes my favourite application of Pascal's Triangle, the Apex Card Trick, although this is also available elsewhere.

As the presenter of this effect, you lay out five cards, face up. From these five cards, predict the apex number and lie it face down before you add the numbers, in the same fashion as Pyramid Problem Solving. Do not let the class know what the predicted number is at this stage. Pick cards from the pack, and lay them out face up to build the triangle up to your selected face down card by adding two numbers to give a single digit answer.

For example, if the first five cards are a 9, 7, Ace, 5 and 8 then the predicted card at the apex that you put face down will be an eight. Where the sum of the two cards is a two digit number, add the digits again.

Example: 9 + 7 = 16, but we do not have a 16 in cards so we add again. We get 1 + 6 = 7, as the first card in the second row.

[ILLUSTRATION OMITTED]

Solution

Write down the first five lines of Pascal's triangle.

[ILLUSTRATION OMITTED]

The numbers in Pascal's triangle tell you the number of paths to the apex for each of the five numbers in the bottom row. To determine the apex number, multiply the value of the first card by one, add 4 times the second card plus 6 times the third card plus 4 times the fifth card plus 1 times the sixth card. Do this calculation, but keep adding the digits in your answer as a running total. Keep going until you get a single digit answer. This method is called 'casting out nines'. You can even cast out nines as you go each time you get a sum of nine or bigger.

In our example,

1 x 9 + 4 x 7 + 6 x 1 + 4 x 5 + 1 x 8 = 9 + 28 + 6 + 20 + 8 = 71 (7 +1) = 8

OR

1 x 9 (= 0) + 4 x 7 (28, 2 + 8 = 10, 1 + 0 = 1) + 6 x 1 (= 6) + 4 x 5 (20, 2 + 0 = 2) + 1 x 8 (= 8) (0 + 1 + 6 + 2 + 8) = 17 = 8

In discussions with your students about doing this mathematical trick you can reinforce ideas about order of operations In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. From the earliest use of mathematical notation, multiplication took precedence over addition, whichever side of a  and mental computation skill, because you need to get the answer correct in your head!

b) A prediction using ten numbers in the base

This can be even more impressive than with the cards. You can ask a volunteer from your class for ten single digit numbers and predict the apex number, writing it up on the board. Here is your starting line starting line
n. Sports
The point or line at which a race begins.

Noun 1. starting line - a line indicating the location of the start of a race or a game
scratch line, scratch, start
 of ten numbers:

8, 1, 3, 5, 6, 3, 2, 1, 7, 9.

Your prediction for the apex number is ...?

Solution

Using ten numbers in the base

[ILLUSTRATION OMITTED]

The tenth line of Pascal's Triangle is:

1, 9, 36, 84, 126, 84, 36, 9, 1

There are only four numbers in the base that do not reduce to zero by the 'casting out nines' process. They are the first (8), fourth (5), seventh (2) and tenth (9) numbers. Note that the number 84 has a remainder of 3 when divided by the number 9, or casting out nines Casting out nines is a sanity check to ensure that hand computations of sums, differences, products, and quotients of integers are correct. By looking at the digital roots of the inputs and outputs, the casting-out-nines method can help one check arithmetic calculations.  gives 8 + 4 = 12, 1 + 2 = 3.

For our example all you need to do to make your apex number prediction is calculate

1 x 8 + 3 x 5 + 3 x 2 + 1 x 9 = 8 +15 + 6 + 9 = 38 (3 + 8 = 11; 1 + 1 = 2)

Therefore number 2 is our apex number.

After working with the apex card trick and the ten number version, the 'problem solving pyramid' and 'cairns', are also more easily understood. When you start with this new set of ten numbers (8, 6, 4, 3, 1, 2, 5, 8, 9, 4) your answer for the apex number should be 9.

(1 x 8 + 3 x 3 + 3 x 5 + 1 x 4 = 8 + 9 + 15 + 4 = 8 + 6 + 4 = 18 = 9 (Do not cast out at the end!)

Extension activities

Investigate Pascal's Triangle further to see how you could find the number at the apex for various sized triangles that have 4, 5, 6, 7, 8 or 9 numbers in the bottom row.

Activity four: Ziggurats

The Babylonians built their temples on the top of stepped pyramids called ziggurats. In examples from Mathematics Cubed: Investigation with Interlocking interlocking /in·ter·lock·ing/ (-lok´ing) closely joined, as by hooks or dovetails; locking into one another.
interlocking Obstetrics A rare complication of vaginal delivery of twins; the 1st
 Cubes (Bramald, Thompson & Straker, 1997, p. 45) you are asked to build some models and develop the correct formulae to go with each of them.

a) Ziggurat ziggurat (zĭg`răt), form of temple common to the Sumerians, Babylonians and Assyrians. The earliest examples date from the end of the 3d millenium B.C.  1

In the bottom layer there are 5 cubes on each side, then 4, then 3, then 2 and then 1 on top.

b) Ziggurat 2

In the bottom layer there are 7 cubes on each side, then 5, then 3 and then 1 on top.

Find formulae for the number of cubes in each layer ([T.sub.n]) and the total number of cubes in each Ziggurat ([S.sub.n]). You could see first if your students can work out the numbers of blocks needed to build these models 4, 5 or even ten high.

Solutions

Ziggurat 1

[T.sub.n] = [n.sup.2]

[S.sub.n] = [n.summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  over (1)][r.sup.2] = [n/6](n + 1)(2n + 1)

Ziggurat 2

[T.sub.n] = [(2n - 1).sup.2]

[S.sub.n] = [n.summation over (1)][(2r - 1).sup.2] = [n/3](2n + 1)(2n - 1)

The reader is left to use the formula, or some other method to work out the numbers needed to build the 4, 5 and 10 high Ziggurats.

With the formulae from the ziggurats and pyramids activities, the teacher is armed with a powerful visual model to demonstrate mathematics from the Sequences and Series topic.

[T.sub.n] can indicate the number of blocks in each layer while [S.sub.n] will be the total number of blocks in each structure of n layers. These formulae can also be extended to develop the method of mathematical induction with a visual representation of [T.sub.n] and [S.sub.n].

Activity five: Pyramidal numbers

In an old book called Math Miracles (Wallace Lee, 1976, p. 20), 'zikkarats' are also mentioned under the heading of 'Some Odd Formulas'. Here are two more examples.

a) A square-based pyramid

A hypothetical pyramid has a bottom layer of stones laid in a perfect square up to one stone in the top layer. If there are 432 stones in each side of the bottom layer how many stones are there altogether?

b) A triangular-based pyramid (tetrahedron tetrahedron: see polyhedron. )

Imagine a pyramid stack of cannon balls. This may be modelled with tennis balls or similar. To make the 'tennis ball pyramids' make a frame similar to those found on a snooker snooker

Variation of English billiards. It is played with 15 red balls and 6 variously coloured balls. Snooker arose, probably in India, as a game for soldiers in the 1870s.
 or pool table to hold the solid together. Find the formula:

i) to give the number of balls in each layer

ii) to give the total number of balls when there are n layers (or n balls on each edge of the pyramid.)

Solutions

a) Square-based pyramid

Using

n(n + 1)(2n + 1)/9

gives us 26 967 240 stones!

b) Triangular-based pyramid

[T.sub.n] = n(n + 1)/2 (triangular numbers (Math.) the series of numbers formed by the successive sums of the terms of an arithmetical progression, of which the first term and the common difference are 1. See Figurate numbers, under Figurate.

See also: Triangular
)

Sn = n(n + 1)(n + 2)/6

These puzzles have all created a lot of interest among students in all my classes, from Years 7-12.

Drawings on large sheets of cardboard were used for group work and A4 worksheets seemed to work best for individual work. The hands-on approach of actually building ziggurats with cubes was also appealing to groups of students.

Use www.askjeeves.com or similar to find out information such as:

* What is a cairn?

* How were the pyramids built?

* How high and how many stones were used in the Pyramids of Giza?

Many of the puzzles presented here are also now on the Internet. I hope that readers find some useful material in these triangles and pyramids and that it leads you to dig up some more motivational material for your students.

Acknowledgements

Thanks to Debbie ('Technology') Tutaki for her patient assistance in helping me with using the technology.

Many thanks also to Dr Jane M. Watson and other members of the AMT See vPro.  Editorial Board for their constructive advice and encouragement.

References

Board of Studies, NSW (1999). Stage 6 General Mathematics Syllabus for Preliminary and HSC Courses. Sydney: Author.

Board of Studies, NSW (2002). Mathematics Syllabus: Years 7-10. Sydney: Author.

Bramald, R., Thompson, I. & Straker, N. (1997). Mathematics Cubed: Investigations with Interlocking Cubes. Colchester: Claire Publications:

Nexus Media Ltd (17 September 1998). Figure it out, No. 6.

Gardner, M. (1975). Mathematical Carnival. London: Penguin Books.

Moodie-Bloom, T. (1995). Brain games: The cairns puzzles. The Australian Magazine, 70.

Wallace, L. W. (1976). Math Miracles. Calgary: Mickey Hades Hades (hā`dēz), in Greek and Roman religion and mythology.

1 The ruler of the underworld: see Pluto.

2 The world of the dead, ruled by Pluto and Persephone, located either underground or in the far west beyond the
 International.

Gary Neville Gary Neville (born 18 February 1975 in Bury, Greater Manchester) is an English footballer who is England's most capped right full back, and Manchester United's club captain.  

Morisset High School, NSW

therealnevilles@bigpond.com
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Author:Neville, Gary
Publication:Australian Mathematics Teacher
Geographic Code:8AUST
Date:Mar 22, 2005
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