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'----"-. . ? \' I ~ . . :.::.--u--' 33
ular nunlbers • • _ .. ,,.,... ............ are and triangular nu
LV') - - - - f'.,...,.._
IITJ 1, 3 and 6 are the first three triangular numbers
.
since they can be arranged in a triangular pattern.
.
Continue this pattern until you have drawn the next three . Ill..
.. . ..
1 + 2 = 3 1+2+3 = 6 10
.. -:
~ y This is one way to write A + A _....
0 Design arrow diagrams like this one 3rd triangular number about the result. .... ~
/\ ml1]1 ~t=-ese to=-make square numbers-...
for the 7th and 8th triangular numbers.
!Ii Without drawing diagrams, a the 3rd and 4th triangular numbers
show how. you can work out 1+2+3 _...,___._ 6 b the 5th and 6th triangular numbers.
the 9th and 10th triangular numbers. I • Choose materials to help you, if you want. Practical: fit together triangular
< numbers to make square numbers
\' • Write about your results.
a Copy and complete this difference pattern Show diagrams for two other square numbers less than 100 to show that:
for the first ten triangular numbers. made from two triangular numbers.
i 3 6 10 @] ~
\/\/\/\/\/\/\/\/\/ Co-=e this table to show ~er totals up to 100. a 6 + 10 = 16 and 16 = 4 2
2 3 '- , -,-~·~-T"" "'"' ,. r'"-, ""--"'+-""-t..., -~·a I
Practical: draw triangular patterns \/ Tnangular numbers -~ 3 ~ 10 15 1 _j._ ___ J __
2
1 b 15 + 21 = 36 and 36 = 6 .
illustrating these numbers. 1 • Square numbers 1 , 4 1 9 1 I I I
·-------·
1 ' 1 + 2 = 3, 1 + 2 + 3 = 6, m What are the positions of the two triangular numbers which together make:
l-.:.._....::..:::...___J __ ..__...r..:_ __ ___j __ .__ a the 19th square number b the 28th square number?
1 + 2 + 3 + 4 = 10,
1 + 2 + 3 + 4 + 5 = 15,
Open: show diagrams for two other
1 +2+3+4+5+6=21
square numbers less than 100
made from two triangular numbers.
Practical: draw arrow diagrams linking these statements.
1 + 2 + 3 + 4 + 5 + 6 + 7 <=>7th triangular number <=> 28
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 <=>8th triangular number <=> 36 Position 1st 2nd 3rd 4th 5th 6th 7th 8th qth 10th
Triangular number 1 3 6 10 15 21 28 36 45 55
Square number 1 4 q 16 25 36 4q 64 81 100
1+2+3+4+5+6+7+8+q=45 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + q + 10 = 55
[!] Challenge

2
a 18th and 1 qth 2 - 1 = 3 Every time you subtract a
a 1 55 2
3 - 3 = 6 triangular number from a
b 27th and 28th 2
4 - 6 = 10 square number which is one
5 2 - 10 = 15 position higher, the answer is
2
6 - 15 = 21 the next consecutive triangular
2
b Blue numbers: the increase between consecutive triangular numbers. 7 - 21 = 28 number.
8 2 - 28 = 36
c Red numbers: the difference in the amount of increase between 2
q - 36 = 45
consecutive triangular numbers. 2
10 - 45 = 55
11 2 - 55 = 66
Challenge
Triangular numbers are built up of layers of consecutive numbers.