Download A Triangular Journey

AC
Triangular Numbers
We are investigating the triangular numbers
We are learning to solve problems in mathematical contexts
We are learning to make generalisation of nth term
EA
AA
AM
Station A – Handshakes
AP
You will need a group of buddies, a piece of paper and a pencil.
This task is all about making a pattern. You are to record how many handshakes
happen when all of you shake hands with each other. Use the table like the one here
to record what you find out.
Number of
people in
the group
1
Number of
handshakes
2
1
3
4
5
6
7
(a) Look at your table and describe here any patterns you can see.
(b) Try and explain why your pattern forms.
(c) How many handshakes can happen between 19 people? ____________________
(d) Who may shake the Queen’s hand? __________________________________
Station B – Intersections
You will need a group of buddies, a piece of paper and a pencil.
This task is all about making a pattern. Each new “road” you make must cross all other
roads. Record the number of intersections that happen as you increase the number of
roads. Here, three roads cross making 3 intersections.
Number of
roads
1
Number of
intersections
0
2
3
3
4
5
6
7
(a) Look at your table and describe here any patterns you can see.
(b) Explain why your pattern forms.
(c) Where do most car accidents happen? _______________________________
Station C – Chords
You will need a group of buddies, a piece of paper and a pencil.
In this task you sketch a circle and place dots around the circumference. Then you
join each dot to the other dots and count up the lines you create. The diagram shows
the lines made with 5 dots.
Number of
dots
1
Number of
joining
lines
0
2
3
4
5
10
6
(a) Look at your table and describe here any patterns you can see.
(b) Explain why your pattern forms.
(c) Which of these patterns can be drawn without taking the pen off the paper?
Why? ___________________________________________________________
Station D – Block Models
You will need a group of buddies, a piece of paper, multilink blocks and a pencil.
In this task you are to make models in the following pattern. Each new model adds one
more column of blocks that is one higher. Add up the number of blocks in each model
and record these in the table below.
Model
Number
1
Number of
blocks
needed
1
2
3
3
6
4
5
6
(a) Look at your table and describe here any patterns you can see.
(b) Explain why your pattern forms.
(c) How many blocks are needed to make the 9th model?
(d) What shapes can two of the same models make?
Station E – N={1, 2, 3, 4, 5, … }
You will need a group of buddies, a piece of paper and a pencil
This task requires some adding or other clever way of to find the total. Your task is
to add the numbers in each row and record the total in the second column.
1
1+2
3
1+2+3
1+2+3+4
1+2+3+4+5
1+2+3+4+5+6
1+2+3+4+5+6+7
1+2+3+4+5+6+7+8
1+2+3+4+5+6+7+8+9
(a) Look at the second column and describe here any patterns you can see.
(b) Explain why your pattern forms.
(c) What do the first 19 whole numbers add up to?
(d) How does the middle number help in calculating the total in each row?
Station F – Bonjour Pascal
You will need a group of buddies, a piece of paper and a pencil.
You are to complete the pattern in the following table. Copy the table, decide upon
the missing values and write them in the gaps.
1
1
1
1
1
2
3
4
1
3
1
1
1
1
(a) Look at the third row and describe here any patterns you can see.
(b) Explain why your pattern forms.
(c) What number must be in the 19th column and 3rd row? _______________
(d) What number is in the 5th row and the 8th column? _________________
(e) Who was Pascal?
Station G – Pyramid
You will need a group of buddies, a piece of paper, match sticks and a pencil.
In this task you can draw or make the pyramid to help generate the answers. Record
the number of triangles in the rows starting from the top row.
Row
Number
Row 1
Rows 1
and 2
Rows 1,2
and 3
Rows
1,2,3 and
4
Rows
1,2,3,4
and 5
Rows
1,2,3,4,5
and 6
Number of
Triangles
1
Number of
Matches
3
3
9
(a) Look at the second column and describe here any patterns you can see.
(b) Explain why your pattern forms.
(c) How many matches are needed to make a pyramid ten rows high? ____________
Exercise 1 – Time to Think Analytically
You will need a group of buddies, a piece of paper and a pencil
In your group look at the results from each of the Stations A to G, or how ever many
you have.
(a) What is it each of these tasks has in common?
Hopefully you will have noticed that each task required you to add the natural
numbers N, starting from 1 each time. Sometimes the first total was zero (Stations
A, B and C) and sometimes the first total was 1.
Algebra Revision • the number after n on a number line is called n + 1. The number
before is called n – 1.
0
1
n-1
n
n+1
(b) Your task now is to generate a rule that connects the top row to the second row in
each of these tables.
Table A • A Rather Forward Looking Rule
1st
2nd
3rd
4th
1
3
6
10
5th
15
6th
21
7th
28
8th
36
Table B • A Rather Backward Looking Rule
1st
2nd
3rd
4th
0
1
3
6
5th
10
6th
15
7th
21
8th
28
Record your considered analytical thinking here.
My Rule for Table A
My Rule for Table B
Exercise 2 – Time to Think Logically
You will need to have done Exercise 1.
Task 1 •The Even Numbers
1st
2nd
3rd
2
4
6
4th
5th
6th
...
nth
(a) Fill in the table above and record anything you notice about the even numbers?
(b) In the cell or gap under the “nth” term write the rule that generates the nth even
number. The rule is ________________
(c) Now we add the even numbers to make a series
1st
2nd
3rd
4th
5th
2
2+4
2+4+6
2+4+6+8
2
6
nth
2+4+6 . . . n
(d) My rule for the sum of the even number series is ______________________
My reason for this is
Task 2 •The Odd Numbers
1st
2nd
3rd
1
3
5
4th
5th
6th
...
nth
(a) Fill in the table above and record anything you notice about the odd numbers?
(b) In the cell or gap under the “nth” term write the rule that generates the nth odd
number. The rule is _________________
(c) Now we add the odd numbers to make a series
1st
2nd
3rd
4th
5th
1
1+3
1+3+5
1+3+5+7
1
4
nth
1+3+5 . . . n
(d) My rule for the sum of the odd number series is ______________________
My reason for this is
Exercise 3 – Multiple Times to Think Logically
You will need to have done Exercise 1 and 2.
Task 3 •The Multiples of Three
1st
2nd
3rd
3
6
9
4th
5th
6th
nth
...
(a) Fill in the table above and record all you notice about the multiples of three?
(b) In the cell or gap under the “nth” term write the rule that generates the nth
multiple of three. The rule is _____________
(c) Now we add the multiples of three to make a series
1st
2nd
3rd
4th
5th
3
3+6
3+6+9
3+6+9+12
3
9
nth
3+6+9 . . . n
(d) My rule for the sum of n multiples of three is ____________________
My reason for this is
Task 4 •The Multiples of Four
1st
2nd
3rd
4
8
12
4th
5th
6th
nth
...
(a) Fill in the table above and record all you notice about the multiples of four?
(b) In the cell or gap under the “nth” term write the rule that generates the nth
multiple of four The rule is ______________
(c) Now we add the multiples of four to make a series
1st
2nd
3rd
4th
5th
4
4+8
4+8+12 4+8+12+16
4
12
nth
4+8+12 . . . n
(d) My rule for the sum of n multiples of four is ____________________
My reason for this is
Exercise 4 – Multiple Multiple Times to Think Logically
You will need to have done Exercise 1 and 2.
Task 5 •The Multiples of Five
1st
2nd
3rd
5
10
15
4th
5th
6th
nth
...
(a) Fill in the table above and record all you notice about the multiples of five?
(b) In the cell or gap under the “nth” term write the rule that generates the nth
multiple of five. The rule is ______________
(c) Now we add them to make a series
1st
2nd
3rd
4th
5
5+10
5+10+15 5+10+15+20
5
15
5th
nth
5+10+15 . . . n
(d) My rule for the sum of n multiples of five is ____________________
My reason for this is
Task 6 •The Multiples of “m”
1st
2nd
3rd
m
2m
3m
4th
5th
6th
...
nth
(a) Fill in the table above and record all you notice about the multiples of m?
(b) In the cell or gap under the “nth” term write the rule that generates the nth
multiple of m. The rule is _________________
(c) Now we add them to make a series
1st
2nd
3rd
4th
m
m+2m
m+2m+3m m+2m+3m+4m
m
3m
5th
nth
m+2m+3m . . . n
(d) My rule for the sum of n multiples of m is ____________________
My reason for this is
Exercise 5 – Parts of a Series • Problem Solving 1
You will need to have done Exercise 1 and 2 at least.
Problem 1
What is the sum of the natural numbers N from 9 to 22?
9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 =
Working
You are not allowed to add these!
Problem 2
What is the sum of the natural numbers N from 100 to 200?
Sum =
Working
If you choose to add these then so be it!
Problem 3
What is the sum of the even numbers E from 10 to 20?
10 + 12 + 14 + 16 + 18 + 20 =
Working (Hint, 10 is the 5th even number)
You are not allowed to add these!
Problem 4
What is the sum of the even numbers E from 100 to 1000?
Sum =
Working (Hint, 100 is the 50th even number)
If you choose to add these then so be it!
Exercise 6 – Parts of a Series • Problem Solving 2
You will need to have done Exercise 1 and 2 at least.
Problem 5
What is the sum of the odd numbers O from 9 to 23?
9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 =
Working (Hint, 17 is the 9th odd number)
You are not allowed to add these!
Problem 6
What is the sum of the odd numbers O between 1000 to 2000?
Sum =
Working
If you choose to add these then so be it!
Problem 7
What is the sum of the multiples of 3 M3 from 30 to 60?
Sum =
Working (Hint, 30 is the 10th multiple of 3)
You are not allowed to add these!
Problem 8
What is the sum of the multiples of 7 M7 between 70 and 700?
Sum =
Working
If you choose to add these then so be it!
Exercise 7 – Proof by Induction
You will need to have done most of the exercises above and found them easy.
Induction is a clever and valuable tool used by mathematicians to prove things. It is
like playing dominos. If the first is true and for some number k the same is true
then we just have to prove that k+1 is true. If that is the case, all are true.
The Process
1. Check that a formula is true for the case n = 1
2. Substitute n = k into the formula
3. Now substitute k + 1 for k and rearrange or otherwise show that the new formula
works or is the same as the old one plus the new term.
4. Since it was true for n = 1, which could have been k, and it is also true for k+1, it is
logical that like the dominoes, our formula is true for all n.
For Example
The formula for adding the natural numbers from 1 to n is ½n(n+1)
1. When n = 1, the sum is ½x1x(1+1) = 1, which we know to be true
2. Let n = k, then the sum of the first k numbers is ½k(k + 1)
3. Now if k = k+1, this formula becomes ½ (k +1)(k + 1 + 1)
= ½(k + 1)(k + 2)
= ½(k2 +2k + 1k + 2)
= ½( k2 + k + 2k + 2)
= ½( k2 + k) + ½(2k + 2)
= ½k(k + 1) + (k + 1)
which of course is the sum of the first k numbers plus the next one, k+1.
4. Q.E.D. [It was true for n = 1, hence it was true for n = 2, and so on]
An Expanation of Q.E.D.
“As a mathematician, I declare that I have established truth of a theorem by
writing at the end of its proof the three letters Q.E.D., an abbreviation for the
latin phrase “quod erat demonstrandum”, which translates as “ what had to be
proved”. On the one hand, Q.E.D. is a synonym for truth and beauty in
mathematics; on the other hand, it represents the seemingly inaccessible side of
this ancient science.”
Extract from page 1, Q.E.D. Beauty in Mathematical Proof, by Burkard Polster
ISBN 0-8027-1431-5, Walker publishing Company, www.walkerbooks.com
Now it is your turn.
(a) Prove by induction that the formula for sum of the n even numbers is n(n+1)
(b) Prove by induction that the formula for the sum of the odd numbers is n2
(c) Prove by induction the sum of the first n multiples of 8 is 4n(n+1)
Good luck!
Exercise 7 – Spreadsheeting.
You will need a computer, a buddy, and spreadsheet software such as Excel.
You task is to use the spreadsheet to make a chart of sequences part of which is
shown below. There are two rules which make this a challenge…
• you must only enter the number 1, once into cell A2
• you must only use one formula in a row but you can copy it.
1
2
3
4
5
6
A
B
Natural Numbers
1
2
Even Numbers
2
4
Odd Numbers
C
D
E
F
G
3
4
5
6
6
8
10
12
H
7
8
Challenge Time
Generate the following sequences
(a) Natural Numbers (see above)
(b) Even Numbers
(c) Odd Numbers
(d) Multiples of 3
(e) Multiples of 4
(f) Multiples of 5
(g) Powers of 2
(h) Toggle ={ 1, -1, 1, -1, 1, -1, 1, -1, …}
(i) {1, -2, 3, -4, 5, -6, 7, …}
(j) 1, 10, 100, 1000, …
(h) 1, .5, .25, .125, .0625, …
There are a great numer of sequences. A very cool website is www.sequences.com
which allows a sequence to be entered and checked against all know sequences.
Usually a history of the sequence is displayed and you very soon realise that someone
else has spent a lot of time figuring out something of which you knew nothing.
A challenge is to come up with a different or unknown sequence. Then test it in the
website above. You never know, you may well become famous!
Good Luck!
Triangular Numbers
Teacher’s Notes
The triangular numbers are one of the most important and valuable series in mathematics. A series
is the sum of a sequence. The sequence of natural numbers 1 + 2 + 3 + 4 + … has the formula
½n(n + 1) where n is the number of numbers being added. This formula is multiplicative and
demands that a student look along the series in an additive way, and across the series at the same
time. This simultaneous thinking of two notions is being multiplicative and explains why many
students do not readily “get” this formula. They are not multiplicative thinkers…yet.
These tasks accelerate the development of the Stage 6 Advanced Adders and give them reason to
use multiplication. There are seven different tasks that can be used as Stations for students to rotate
through in small groups and collaborate to answer the questions. Each rotation would take about 10
minutes (or as long as needed). Then for Exercise 1 the teacher brings the students together and
with quietish thinking time has them construct the rules ½n(n + 1) and ½(n - 1)n.
The use of these rules is opportunity for further development. Students who need to confirm the
rules should return to each exercise and check it out. Further development is applying the formula
to the even number series (by doubling), the odd number series (by then subtracting n), the
multiplies of 3, 4, 5…n (by multiplying accordingly), and “chunks” of the series by using
difference.
For experts and extension there are some problems using the sum of the sum of the natural numbers.
See Mathematical Investigations SNP www.nzmaths.co.nz (Pyramids of Balls) and using the skill
of Proof by Induction formulas for the sum of the squares, cubes and so on can be generated.
These exercises are designed to be used in a class with rotating groups of 3, 4 or 5 students. Some
of the tasks might require groups to join up to make more people if needed. The first 5 tasks form
stations. Typically about 10 minutes is needed at each station. Students rotate through these and
then all proceed to Exercise 1 where hopefully sense is made of the findings.
Number Framework domain and stage:
Multiplication and Division – Advanced Additive, Advanced Multiplication
Curriculum reference:
Number Level 4
Materials:
• multilink blocks
• rulers or strings
Diagnostic Questions:
• What is the sum of the first 19 whole numbers?
• What is the sum of the first 99 whole numbers?
• What is the rule for adding n whole numbers?
A student who can answer all these has nothing extra to learn from the tasks 1 to 6 and should
attempt to solve the problems in tasks 7 and 8.
During these activities students will meet:
• nth rule
• multiplication
• patterns
Background:
Further information to back up what they will meet:
• FIO Books – Check Number Books for doubling activities.
• Digistore Activities – See http://www.nzmaths.co.nz
• National Archive of Virtual Manipulatives (via Google)
Question comments and solutions to these exercises
The overall aim of this unit of work is to generate the formula for adding the whole numbers which
is ½n(n+1). This calculates the nth triangular number. This pattern should be common knowledge
soon after students become multiplicative as it appears in many problems. From this pattern the
general term for the even numbers, odd numbers, multiples, squares and cubes, groups of whole,
even, odd and multiple numbers, squares and cubes can be formed. The connection with Pascal’s
Triangle should also be established and in this unit an opportunity to use a spreadsheet to generate
sequences and series is presented. The spreadsheet can be additive or a recursive calculation or a
multiplicative or rule driven sytem of equations.
The Exercises 1 to 7 are tasks that should be set up at stations which the students spend about 5
minutes at to work through. Vary this according to the ability of the students.
Station A • Handshakes
This task is the famous handshake problem which generates the pattern of triangular numbers 0, 1,
3, 6, 10, … as the number of people in the group increases from 1. Some guidance may be needed
to establish a handshake as one interaction between two people. For example, Manu shaking hands
with Fred is the same handshake as Fred shaking hands with Manu.
(a) A common answer here is that the pattern increases by 1 each time. Firstly 1 is added, then 2
then 3 then 4 and so on. This is a typical “additive” response. The student is looking down the
pattern only and not across at the same time. These are the target group of students to move.
Multiplicative students present will offer a different view.
(b) A pattern students may report is as an extra person is added to the group so the number of total
handshakes increases by 1 less than the number of people now in the group. For example the 4th
person in the group adds 3 more handshakes. This is a recursive description. Another way of saying
this is “every extra person has to shakehands with every one else”.
(c) 19 people can form 1 + 2 + …18 + 19 handshakes. This is 171 = 19x18/2
(d) No one shakes hands with the Queen! The Queen may choose to shake hands with whosoever
she so chooses.
Station B • Intersections
This nice problem repeats the result from Exercise 1 perfectly. The astute student may notice this.
Connecting one problem to another is only to be encouraged. It is more obvious here that an extra
road crosses all previous roads.
(a) The pattern iin the numbers goes up one more each time. We add 1, then 2, then three and so on.
(b) Each new road crosses all previous roads and so the pattern is generated. The 4th crossed 3, the
3rd crossed 2, the 2nd crossed 1 and the 1st did not cross any road.
(c) Most car accidents happen at intersections.
Station C • Chords
This nice geometrical pattern generates the same pattern as exercise 1 and 2. Students should notice
this fact! The first three exercises are toplogically equivalent problems.
(a) The numbers go up by the counting numbers. Perhaps we may see evidence of looking across
where a student might say “the two row numbers add to give the next down number”.
(b) The 5th dot joins the 4 others, the 4th joins 3 others and so on. A multiplicative answer might be
“the 5 dots all join 4 others, but each is counted twice when we do this.
(c) This is a good topographical problem. The 2 dot and 3 dot circles can be drawn because they
have less than or exactly two odd nodes. The 4 dot pattern has 4 odd nodes and so there is 2 nodes
where you can visit but never leave. The maximum number of nodes for a pattern to be traversible
is 2; a start and a finish.
Station D • Block Models
Multi-link blocks have to be an invaluable aid to every classroom. Here they are used to make the
triangular numbers and it is obvious that they form the 1 + 2 + 3 pattern and are the sum of the
natural or whole numbers. Make these models and understand this idea. It is important these models
are made by students.
(a) The pattern goes up by the counting numbers 1, 2, 3, 4
(b) The pattern forms because we add the next highest number as a column.
(c) The 9th model will need 1 + 2 + …8 + 9 blocks. This is middle number 5 x 9 the number of
blocks = 45. That is a multiplicative answer. An additive answer is 1+9 = 10, 2+8 = 10, 3+7 = 10, 4
+ 6 = 10 and 5 makes 45.
(d) There are various shapes that can be made. A rectangle, a staircase, a “jaw” and a 3d corner.
Station E • N={1,2,3,4,5, …}
At last the series of natural numbers is made obvious. This acvtivity moves the model to the
imaging stage of the numeracy teaching model. The generalisation will happen when the teacher
brings everyone together if not before in individual or group cases. Encourage the generalisation as
this is what we all need to live and think effectively.
Station F • Bonjour Pascal
Pascals Triangle is very famous and is generated by addition of the cells above and in this case to
the left. The original and usual triangle is an inverted V. The spreadsheet is not shaped that way so
it becomes rectanglar. The formula is the same and all results are identical.
In this exercise just the simple pattern of the sum of the counting numbers is extracted.
(a) The pattern is 1, 3, 6, 10, 15, …
(b) This is the addition of the whole numbers 1, 1+2, 1+2+3 and so on.
(c) The 19th column,m 3rd row must contatin the number 190.
(d) This answer is found by perseverance! Answer is
(e) Pascal was a rather brilliant Frenchman who lived in the 18th Century. He developed a lot of the
theories of probability because he was employed to exploit his brains against the gambling games of
the day. This he excelled in and was paid a lot of money. Check this out by Googling his name.
Station G • Pyramid
This exercise is about triabgles and is another example of the addition of the whole or natural
numbers.
(a) The pattern is the same 1, 3, 6, 10, …
(b) The pattern forms because in every following row the number of triangles increases by 1 in the
same wayteh natural numbers increase.
(c) Te answer is equuivalent to the sum of the first 10 counting numbers. Ans = 55.
Exercise 1 • Time to Think Analytically
Here it is expected the teacher to ask the students to reflect and consider their findings, and report
any generalisations and observations for the collective group to think about. The key competencies
of participating and contributing come to the fore alongside communication in the form of speaking
and listening. This is a great opportunity to create that “zone of ambiguity” that Vigotsky talked
about which causes learning. Most students in Year 9 generate a number sequence by moving from
one to the next in a recursive way. This is an additive and very tiem consuming and potentially
restrictive way of thinking or positioning oneself. It is a much better and more powerful strategy to
think of the term number or position and the value at the same time. This is being multiplicative
and will reap rewards beyond imagination in both the short term and the long. It is the teacher’s task
to make every student a multipicative thinker and this is a perfect opportunity to do so.
(a) Each of the tasks has in common the pattern 1, 3, 6, 10, …
(b) Table A: The rule required is in words “the value below is the term above times the next term all
divided by 2”. In algebra this is n (n + 1) /2. Any valid version fo this is correct.
Table B: The same idea only in words “ the value below is the term above times the term before all
divided by 2. In algebra this is equivalent to n ( n – 1) /2. Again any valid version is OK.
The analytical thinking here is that students are “seeing” across and along at the same time. This
simultaneous thinking of two different things is the key to mutliplicative thinking. Do it and win!
Exercise 2 • Time to Think Logically
Having established the rule for adding whole numbers we can use this to generate other rules. Here
the rule for even numbers is established.
Task 1 • The Even Numbers
(a) One must notice that the even numbers are just double or twice the counting numbers. This
simple observation is easily overlooked as useful.
(b) This question is engineered to help extract the observation mentioned in (a). the answers here
are obvious but the rule is often not that clear at all.
The rule for the even numbers is 2n. Any equivalent statement is OK. Even in words such as “ the
answer is just double the term number”. Never penalise the thinking for not knowing the syntax of
algebra. The algebra language will develop as needed and with time. It is more important that the
concept of “twice the term number” is established.
(d) A series is the sequential (from the 1st term) sum of a sequence. The pattern here is 2, 6, 12, 20,
and so on. The last term is quite hard to see but is the “term x the next term) or n(n + 1). It will not
be obvious to students but it is simply “double” the rule for adding the counting numbers. The
reason for this is that the even numbers are just double the counting or natural or whole numbers.
Task 2 • The Odd Numbers
This is another opportuntiy for students to connect one problem with another.
(a) The odd number series is of course 1, 3, 5, 7, 9, … An observation may be that it is all the
numbers between the even numbers. Another may be that the odd numbers are just one less that the
even numbers. Any observation like this is good because it connects previous problems to this one.
(b)The nth rule for odd numbers is in words “ twice the term -1” or in algebra 2n – 1. See comments
for even numbers (b).
(c) The series is a very powerful pattern (excuse the pun). It is the square numbers 1, 4, 9, 16, 25 …
(d) The nth term could be simply” term x term” or n2. Another version is “it is the same as the even
numbers but n less”. In algebra is n(n+1) – n which creates an algebra problem to show this is the
same as n2. Either result with supporting reasons is satisfactory.
Exercise 3 – Multiple Times to Think Logically
These exercises practice and push the connection of the formula for the sum of the natural numbers
to another contact. It was used for the even numbers, the odd numbers and now the multiples of 3.
Task 3 • The Multiples of Three
(a) The table becomes 3, 6, 9, 12, 15, 18 and so on. In describing this pattern it is very very
important that students who see the pattern as “just add 3 Sir!” are directed to look again. The
multiplicative answer here is “three times the term number”. Do not reinforce additive thinking at
Year 9. Expect and demand multiplicative thinking. Fail to do this and lament your future
examination results and, worse, damn your student to a life of addition addiction.
(b) The expected nth rule is 3n or “three times the term number”.
(c) The series is 3, 9, 18, 30, …
(d) the rule for the nth term is 3 x the rule for the natural numbers, or 3n(n + 1)/2
Task 4 • The Multiples of Four
This exercise is for those who need more practice and have not already generalised to all multiples.
(a) 4, 8, 12, 16, …
(b) 4n
(c) 4, 12, 24, 40, …
(d) 4n(n + 1)/2 = 2n(n + 1). The reason expected is the answer is just 4 times the formula for the
natural numbers because the multiples of 4 are four times as big as each of those numbers.
Exercise 4 – Multiple Multiple Times to Think Logically
Task 5 • The Multiples of Five
This exercise is for those who need still more practice and have not already generalised to all
multiples. The pattern can be extended to the multiples of 6,7,8,9 etc as required.
(a) They go up in fives, or they are 5x the counting numbers.
(b) The nth term is 5n
(c) 5, 15, 30, 50,…
(d) Rule is 5n(n + 1) /2. The rule is 5 times the rule for the natural numbers.
Task 6 • The Multiples of “m”
The generalisation to deduce a rule for adding the multiples of any number is powerful. This
generalisation can be used to develop further generalisations.
(a) The values go up in “m’s” or are m times the rule for the counting numbers.
(b) Rule is mn
(c) m, 3m, 6m, 10m, …
(d) The rule for the nth term of the series is mn(n + 1)/2. The reason is that the sum is m times the
rule for the counting numbers.
Exercise 5 – Parts of a Series • Problem Solving 1
The exercises here are set up as problems and can be made into stations like the first part of the unit.
Students would attend to the problem and then move as required. The problems are all based on the
previous work using the formula n( n + 1)
Problem 1
One way to solve this problem is subtract the sum to 8 numbers from the sum to 22 numbers. The
answer then is 22(22 + 1)/2 – 8(8 + 1)/2 = 217. This can be confirmed by adding the numbers from
9 to 22. The answer 217 can also be confirmed by the product of 15.5 and 14, the middle number
times the number of terms. Adding here as a confirmation is OK but encouraging the multiplicative
way of checking is advised.
Problem 2
This is an identical problem to Problem 1. Adding this to find the solution would be a very tedious
undertaking. The answer is sum to 200 less the sum to 99. Or 200x201/2 – 99x100/2 = 15150. This
can be checked by 150 x 101 which is the middle times the number of numbers. Adding as
mentioned also works.
Problem 3
The important understanding here is that given in the hint. The answer is then the 10th even sum less
the 4th even sum. The calculation is 10(10+1) – 4(4+1) = 90. This is easily c hecked by adding or
the product 15 x 5.
Problem 4
The answer is 248050. See above for ways to solve this.
Exercise 6 – Parts of a Series • Problem Solving 1
The exercises