AC
Thinking Inside the Square
EA
We are developing thinking skills of connecting and analysing
We are exploring some properties of square numbers
We are developing number strategies using square numbers
We are practising the communication of explanation
AA
AM
AP
Exercise 1 – A Square Number
You may not have a calculator but you can have a friend.
This exercise is to make sure you know all the basic square numbers.
The Idea
A number multiplied by itself makes a square number.
We should understand
3 x 3 = 9 where 9 is the square number and what it looks like.
The number 9 is the square of the number 3.
We can also write 9 = 32 and read this as “9 equals 3 squared”.
We can also say “3 squared equals 9” because that is how the “=” sign works.
Multiply these
1)
2x2
(2)
5x5
(3)
1x1
4)
4x4
(5)
9x9
(6)
10 x 10
7)
7x7
(8)
6x6
(9)
8x8
Square these numbers
10)
7
(11)
8
(12)
9
13)
6
(14)
5
(15)
2
16)
4
(17)
0
(18)
1
Task of the century
On a 100 board put counters on all the square numbers. Do you notice any patterns?
Write the squares of these numbers underneath.
1
2
3
4
5
6
7
8
Write here any odd thing you have you noticed. Explain your observation.
9
Exercise 2 – An Associative Problem
You will need a friend and you will not need a calculator.
This exercise will help you know some the squares of the decades to 100 and uses
your knowledge of the basic squares.
Place Value Revision
You should know that 20 = 2 x 10. That is what two-tens or two-ty or twenty means.
Expand these numbers in the same way. Eg 80 = 8 x 10
1)
30
(2)
60
(3)
70
4)
50
(5)
90
(6)
10
7)
40
(8)
20
(9)
80
You may not know how to multiply 20 x 20 and get 400, but here is one way.
20 x 20 --> 2 x 10 x 2 x 10 --> 2 x 2 x 10 x 10 --> 4 x 100 --> 400
We collect the 2’s and the 10’s together. Why?
400 is the square of 20
20 squared is 400
202 = 400
Square these numbers.
10)
30
(11)
60
(12)
70
13)
50
(14)
90
(15)
10
16)
40
(17)
20
(18)
80
Use the same expanded numeral idea to square these numbers
19)
300
(20) 600
(21)
700
22)
500
(23)
900
(24)
100
25)
400
(26)
200
(27)
800
And lastly these really big numbers.
28)
3000
(29) 60,000
(30)
700,000
31)
10,000,000
(32)
50,000,000
(33)
700,000,000
34)
a billion
(35)
a quadrillion
(36)
a googol
What happens to the number of zeros (place-holders) when the number is squared?
Exercise 3 – A Distribution Problem
You will need a friend.
This exercise will help you know and remember some of the squares bigger than 10.
25 x 25 = (20 + 5) x (20 + 5)
but what does this equal?
Why did we choose (20 + 5) and not something else like (17 + 8) which also equals 25?
The array model of multiplication shows what happens and how to get the answer.
X
20
5
20
20x20=400
100
5
100
25
The answer to 25 x 25 is equal to 400 + 2 lots of 100 + 25 = 625.
Explain where each of the numbers comes from to a friend.
Now use this model to square all of these numbers.
1)
15 = (10 + 5)
(2)
12 = (10 + 2)
(3)
17 = (10 + 7)
4)
11
(5)
14
(6)
19
7)
18
(8)
16
(9)
15
10)
13
(11)
21
(12)
45
What do you notice in these answers?
Write the squares of these numbers underneath.
11
12
13
14
15
16
17
Invent a game that will help you to memorise these square numbers.
18
19
Exercise 4 – A Square is an Odd Combination
You will need some colourful mult-ilink blocks and a cololurful friend.
The odd numbers are quite closely related to the square numbers. This exercise
explores this interesting relationship. It is a very useful to have this understanding.
Firstly make a 1 square then add an L shape of 3 in a different colour. Then add L5
and so on always using a different colour.
What do you notice?
Apart from the lovely colours and having fun with the blocks; it is interesting to
notice that the odd numbers add up to make the square numbers.
1 + 3 = 22
1 + 3 + 5 = 32
1 + 3 + 5 + 7 = 42
What do you notice in this pattern?
What do you think the first 5 odd numbers add up to? Explain…and check.
What would the first 10 odd numbers add up to? Explain…and check.
Can you generalise this idea and say what any number of odd numbers would sum to?
Exercise 5 – Odd Patterns.
You will need some colourful multilink blocks and yet another colourful friend.
This exercise shows how to explore the way odd numbers combine.
The set of Odd Numbers = { 1, 3 ,5, 7, 9, 11, 13, 15, 17, 19, 21, 23, …}
There is not enough paper in the universe to write this set out completely. In fact
there is not enough ink either. Nor is the universe large enough to stuff all the paper
you would need. The odd number set is an example of an infinite set.
Fill the gaps and extend this pattern.
Summing the Odds
1
1+3
1+3+5
1+3+5+7
Sum
1
4
A Pattern
12
22
32
16
36
What do you notice?
Another way to look at the odd numbers. Fill in the gaps and extend the pattern
Curious Odd Numbers
Sum
A Cube Pattern
1
1
1 x 1 x 1 = 13
3+5
8
27
3 x 3 x 3 = 33
13 + 15 + 17 + 19
4 x 4 x 4 = 43
In this pattern what is the middle number in the 3rd row?
What is the middle number in the 11th row?
What do the numbers in the 5th row add up to?
What do the numbers in the 11th row add up to?
Now explore and see if you can discover an odd pattern.
Exercise 6 – Square Stepping Stones
You may need some colourful multilink blocks and a colourful buddy.
Here is a problem to solve.
If you know that 12 x 12 = 144,
how can you figure out what the answer to 13 x 13 is?
To solve this we will use the “make it smaller” strategy.
If you know that 3 x 3 = 9, how can you know what 4x4 is?
Here is 3 x 3. To make 4 x 4 from the 3x3 I copied a
column of 3 and pasted them as “red” smiley faces. Then
I copied a row of 4 and pasted them as “aqua” smiley
faces.
Looking from left to right;
I added a red face to each row and then a row of 4.
4x4
=3x3+3x1+1x4
= 3 lots of 3 and 3 lots of 1 and 1 lot of 4
= 3 lots of 4 and 1 lot of 4
= 4 lots of 4
=4x4
Another way of getting to the answer is add two more
lots of 3 and then 1 for the missing corner.
Can you see any other ways to make the 4 x 4?
Use one of these methods to find all the squares to 23 starting with the 9 x 9 = 81.
1
2
3
4
5
9x9
10 x 10
11 x 11
12 x 12
13 x 13
81
6
7
8
9
10
14 x 14
15 x 15
16 x 16
17 x 17
18 x 18
11
12
13
14
15
19 x 19
20 x 20
21 x 21
22 x 22
23 x 23
If 32 x 32 = 1024, what is 33 x 33? _______________________________
If 222 x 222 = 49284, what is 223 x 223? ___________________________
Exercise 7 – A Square is Triangular!
You will need some colourful multilink blocks and yet another colourful buddy.
This exercise explores the way triangular numbers combine and make the squares.
Firstly triangular numbers 1
3
6
10
15
… … …
Why do you think they are called triangular numbers?
Above are models for the first 5 triangular numbers.
We will call them T1, T2, T3, T4 and T5. Note that T3 = 1 + 2 + 3 = 6 blocks.
What does T8 = _______________________ ?
What does T10 = _______________________ ?
Now for an interesting twist.
If we take T2 and turn it over, it fits into T3 and makes a 3 x 3 square.
Make models using the multilink blocks and explore what happens when two
consecutive triangular numbers are combined in this way.
Record here what you found out.
Check using the numbers above that two consecutive triangular numbers make a
square number and write the results here.
What two triangular numbers make up the square number 12 x 12? _____ _______
What two triangular numbers make up the square number 289 ? _______ _______
What two triangular numbers make up the square number 10,000 ? ______ ______
Exercise 8 – Connecting Squares, Odds and Triangular
You will need an odd friend to explore this puzzle and some colourful multilink blocks.
This exercise shows how ideas in mathematics are connected.
The Connection
A square is the sum of some odd numbers. Eg 32 = 9 = 1 + 3 + 5
Two triangular numbers join to make a square number. Eg T2 + T3 = 3 + 6 = 9 = 32
So the odd numbers must be connected to the triangular numbers.
How are they connected?
A
B
The connection
is a round and
round story
C
Study the diagrams above. Look for links from one to the next. Go around clockwise
and then the other way.
Now do it! Make a set of these with the multilink blocks and transform them from
one shape to the next.
At A you can see how 1 + 3 + 5 join together to form the 3 x 3 square.
At B you can see how the square is made up from the T2 and T3 triangular numbers.
Look carefully at C and find the link between T2 and T3 and the odd numbers.
Write here what you see.
The mind shift you need to do is to look at the shape in a
different way. Look across the layers and see the 1 in the
top layer, the 3 in the next layer and then the 5 in the
bottom layer. This is an important problem solving technique to remember and use.
Last thought. The odd numbers are closely related to the even numbers! Hmmmm.
Exercise 9 – Squaring Down
You may need some colourful multilink blocks and a yet another colourful buddy.
Here is another problem to solve.
If you know that 13 x 13 = 169,
how can you figure out what the answer to 12 x 12 is?
To solve this we will us the “make it smaller” problem solving strategy.
If you know that 4x4 = 16, how can you know what 3x3 is?
Here is 4 x 4. To make 3 x 3 from the 4 x4 I take away
the “aqua” row of 4 and then the “red” column of 3.
I subtracted a row of 4 and then a column of 3.
4x4
=3x3+3+4
4x4-4
= 3 lots of 3 and 3
4 x 4 – 4 - 3 = 3 lots of 3
=3x3
Another way of getting to the answer is subtract two
lots of 3 and then subtract the lonely 1.
Why do we always subtract an odd and an even number?
Use any method to find these squares starting with the 30 x 30 = 900.
1
2
30 x 30
29 x 29
900
5
6
70 x 70
69 x 69
4900
9
10
100 x 100
99 x 99
10,000
3
4
25 x 25
24 x 24
625
7
8
75 x 75
74 x 74
5625
11
12
250 x 250
249 x 249
62500
If 50 x 50 = 2500, what is 49 x 49? _______________________________
If 1000 x 1000 = 1,000,000, what is 999 x 999? _______________________
If 13 x 13 = 169, what is 12 x 12 ? ________________________________
Exercise 10 – Squares that Differ by Two
You may need some colourful multilink blocks and a really colourful buddy.
Here is a bigger problem to solve.
If you know that 12 x 12 = 144,
how can you figure out what the answer to 14 x 14 is?
To solve this we will us the “make it smaller” strategy again.
If you know that 3 x 3 = 9, how can you know what 5 x 5 is?
Here is 3 x 3. To make 5 x 5 from the 3 x 3 I copied two
columns of 3 and pasted them as “red” smiley faces. Then
I copied two rows of 5 and pasted them as “aqua.
Looking at this across the faces or left to right.
I added two “red” smileys to each of the 3 rows and then
added two rows of 5.
5x5
=3x3+3x2+2x5
= 3 lots of 3 and 3 lots of 2 and 2 lots of 5
= 3 lots of 5 and 2 lots of 5
= 5 lots of 5
Another way of getting to the answer is add two more
green faces for each row and for each column and then
2 x 2 for the missing corner.
There are other ways to look at this problem.
Use one of these methods to increase all these squares.
1 9x9
2 11 x 11
81
5 42 x 42
6 44 x 44
1764
9 89 x 89
10 91 x 91
7921
3 22 x 22
4 24 x 24
484
7 51 x 51
8 53 x 53
2809
11 120 x 120
12 122 x 122
14400
12 x 12 = 144, so what is 14 x 14? _______________________________
222 x 222 = 49284, so what is 224 x 224? ___________________________
What could you do if the increase was more? Eg 20 x 20 = 400 so what is 27 x 27?
Exercise 11 – A Curious Square Trick
You will need a secret friend
65 x 65 = 4225
We can calculate the answer to similar
numbers like 35, 45, 95 very easily.
It always ends in 25
The hundreds digits are 6 x (6+1) = 42
Joining these gives 4225
Work out these using the rule in the box.
1. 25 x 25
2. 35 x 35
4. 75 x 75
5. 15 x 15
3. 55 x 55
6. 85 x 85
It even works on bigger numbers
165 x 165
Here the hundreds is
16 x 17 = 16 x 16 + 16= 282
Answer is 28225
But we do need to know how to multiple
consecutive numbers like 16x17
Work out these big squares
7. 105 x 105
10. 155 x 155
8. 115 x 115
11. 185 x 185
9. 125 x 125
12. 195 x 195
But wait…why does it work?
So far we are only doing numbers.
We are not doing mathematics until we know
why it works;
And can explain it to someone.
Investigate this problem and write your thoughtful answer to explain why it works.
Exercise 12 – Introducing the Square Root
You will need a very secret friend and a calculator.
This exercise is an introduction to the “inverse” of squaring a number.
The Basic Idea
12 x 12 = 144
Here the 144 is the square of 12
But we can look at this another way.
Here the 12 is the square root of 144.
Why it is called a square root is another puzzle!
A square root is the number which when you multiply it by itself will give you the
original number!
An example is 9 which has a square root of 3, because 3 x 3 = 9.
Arranging the number 9 in a square pattern illustrates where
the 3 comes from. It is the length of one side of the square.
How about the number 10?
Can you think of a number that when you square it will give you 10?
How about the number 16?
Can you think of a number that when you square it will give you 16?
√
Sometimes it is very easy to get the square root but usually it is very difficult. That
is one good reason why there is a “square root” button on every calculator.
Estimate the square root answers to these problems. Some easy, some hard.
1. 25
2. 24
3. 26
4. 64
5. 60
6. 70
Use your calculator to find the exact answers.
Now for a bewildering insight. See if you can explain this!
Estimate the square root answers to these problems. Some easy, some hard.
7. 250
8. 240
9. 260
10. 640
11. 600
12. 700
Use your calculator to find the exact answers.
Why were these answers not related to the other 6 problems?
Exercise 13 – Revision of “n”
You will need a long paper and a pencil.
Draw an empty number line like this
Now mark the number 0 or zero near the middle.
Now place marks for all of these numbers.
(a) n
(b) the number one less than n
(c) the number 3 less than n
(d) the number 5 less than n
(e) the number 4 more than n
(f) the number 1 bigger than n
(g) the number 2 more than n
(h) the number halfway between 0 and n +1
(i) the number 2
(j) the number 2n
(k) the number 2n +2
(l) the number 2n + 2n
(m) the number that is ¼of n smaller than n
(n) –n
(o) -1
(p) 1 - n
(q) –n - 1
(r) –n + 1
(s) -n + 2
(t) -2n
(u) n2
(v) n3
(w) square root of n
(x) 1 nth
(y) n nths
(z) the sum of n numbers
Thinking Inside the Square
Teacher’s Notes
These exercises, activities and learning activities are designed for students to use independently or
in small groups to practise number properties. Some involve investigation (Mikes Investigation
Sheet link) and may become longer and more involved tasks with subsequent recording/reporting.
Typically an exercise is a 10 to 15 minute activity. The exercises can also be used as teaching
activities with similar supporting work for students.
Number Framework domain and stage:
Multiplication and Division – Early Additive, Advanced Additive, Advanced Multiplication
Curriculum reference:
Number Level 3 and 4
Numeracy Project book reference:
These exercises and activities follow from a teaching episode based on Book 6. Extra teaching
episodes are described with the exercise notes below.
Materials:
• Multilink blocks
• 100s chart and flipblock 100s board
• counters
• calculator
Prior Knowledge. Students should be able to:
• explain a square as 3 x 3 = 9 and model this idea.
• be developing the square tables.
• be able to recognise the odd, even, square and triangualr numbers
• be able to work with larger numbers
• know of the array model of multiplication
• be developing notions of meaning for expressions such as n+2
During these activities students will meet:
• placevalue of whole numbers
• square, odd, even, triangular and other sets of numbers
• patterns of numbers
• commutative, distribution and associative properties
• square roots of numbers
Background:
Further information to back up what students will meet:
• FIO Books – Check Number Books for square activities.
• Digistore Activities – See http://www.nzmaths.co.nz
• National Archive of Virtual Manipulatives (via Google)
Comments on these exercises
These exercises are powerful illustrations of the connectivity of mathematics. The square of number
is established and then the odd numbers are modelled andn linked to the squares. The triangualr
numbers are modelled and the squares are linked to these. All the while patterns are being sort and
larger numbers with repetition of the smaller squares reinforced. The link of odds to squares and
squares to triangualr therefore odds to triangualrs is discovered. This is a form of the equivalence
relationship or the Zeroth Law of Thermodynamics (If A= B and B=C then A=C). The exercises
then develop into using the square knowledge. If yuou know 12x12 then what do you know about
13x13? While it could develop to the generalisation of (x-a)(x-b) this is not done.
Exercise 1
Teaching lessons preceeding this exercise could include:• revising multiplication tables and patterns
• establishing the notion of square by modelling
• locating the square numbers on a 100s board
• looking for patterns of numbers
In this exercise the notion of square is established with the common single digit squares being
practiced. Locating these on a 100s board should evoke the odd number pattern between the
squares. It is important different visualisations of these numbers are given. The reverse notion of
square root is not mentioned but could be as an advanced organiser for exercise 12. Revise the way
an equals works both ways and what it means. It does not mean “evaluate this”.
Exercise 2
Teaching lessons preceeding this exercise could include:• use of place value to expand numbers
• how we can “part-whole” numbers and associate them as we choose
• really big numbers and names of them
The way 20 x 20 becomes 2x10x2x10 = 2x2x10x10 is very important. This needs to be explored
and established as knowledge. There is a lot of language surrounding the use of square.
The “quad” on quadrillon means 4 groups of 000 before the million so 1 quadrillion is
1,000,000,000,000,ooo,ooo and is the same as 1000 trillion (tri meaning 3) and 1000,000 billion (bi
meaning 2). A googol is from Charlie Brown cartoon series in which Schrodinger, in response to a
question from Lucy, estimates the chances that he is in love with her as 1 in a googol or extremely
slim. A googol is 10 to the power of 100 or 10100. It is fun to write this number out. What is a
googolplex?
Exercise 3
Teaching lessons preceeding this exercise could include:• developing the array model as a way to multiply larger numbers
• linking the array model to the common algorithm for multiplication
• matching squares through “memory” or “pickup” games.
The array model is an extremely important as a way to solve problems. It uses the area notion and is
difficult for additive students to comprehend. It is this notion that is a barrier for these students. Use
it with oodles of explanation. Repeat the use of the word “lots” to establish it is groups that are
being worked. A challenge is for students to invent their own games an so develop responsibility for
their own learning.
Repeat! Additive students will find areas and squares very challenging.
Exercise 4
Teaching lessons preceeding this exercise could include:• use of models to illstrate odd, even, triagular, multiples of numbers
• use of colour to stress patterns
• development of the idea of generalisation leading to algebraic notions
This property of odd numbers is a very useful and commonly occuring pattern. It is beautifully
modelled as a L and joined as a square when summed from 1. Summing any consecutive odds leads
to the difference of two squares quite easily but should be reserved for the better student or later
when it can be explored properly and fully. Other models of odd numbers using side by side shapes
can be useful as well to illustrate even and oddness.
Exercise 5
Teaching lessons preceeding this exercise could include:• patterns in numbers starting from 1
• freedom and confidence to explore and seek out patterns
• the notion of infinity
The odd number set can be ordered in many ways. Two of these are shown and are very useful and
very interesting respectively. The squares come from summing the odds. The cubes and squares
come from ordering them in a counting sort of way. Patterns within patterns. All of this is good
development of thinking.
Exercise 6
Teaching lessons preceeding this exercise could include:• the connecting of one square to the next by seeing the added parts
• generating a pattern
The powerful question “if you know that 12x12=144, what do you know about 13x13” promotes
investigation and has several obvious answers. The answer can be additive or multiplicative or a
combiination of the two. All options should be discovered and made available to students. Using
this idea is import as an example of what can be done. This carves a problem solving pathway
which will prove useful in later mathematics. The use of the word “lot” and how they are grouped is
interesting and illustrates the distributive law first by the common term and then commutatively by
the common group.
Exercise 7
Teaching lessons preceeding this exercise could include:• modeling of counting numbers and summing these to make triangualr numbers
• solving the common problems of sum the first 10 counting numbers in various ways
• sum to 100, 1000 to and development of the formula n(n+1)/2 for better students
• It is very important to explore and know how to sum the triangular numbers
Use of colours highlights the count. This can be done with all students. What they will see is limited
by their numeracy stage. A counter will see just that. An adder will see patterns and may be able to
make groups of equal size. A grouper will see the area model and may be able to generalise the
solution. The consecutive idea of two T numbers Tn-1 and Tn making a square of nxn is only on eof
many such patters and others can be explored. Knowing the algebra for these sums can lead to
iinteresting proof by algebraic manipulation. Eg Tn = n(n+1)/2, show Tn-1 + Tn = nxn.
A curious extension is asking “Are any triangular numbers also square numbers? The answer to
this is left for the curious to consider.
Exercise 8
Teaching lessons preceeding this exercise could include:• rearranging triangular numbers to make different patterns.
• showing the logic of if A=B and B=C then A = C using temperature of three joined metal
containers.
The important link here is that students see that of the odds are connected to the squares and the
squares are connected to the triangualr numbers then the odds MUST ALSO be connected to the
triangular numbers. Allow time for students to find out the connection. It is a mind shift and
illustrates another important problem solving strategy…look at the problem another way. An
extension to more connections and the infinty of such connections is suggested as a last thought.
Exercise 9
Teaching lessons preceeding this exercise could include:• using a model to see a new problem.
This is the reverse of the previous exercise 6 to point out another way to look at a problem. It is
applied in some useful ways.
Exercise 10
Teaching lessons preceeding this exercise could include:• modelling of the problem and looking for patterns
• different ways to look at the problem.
This exercise is an extension to a greater difference up (and down) from a know square number. It
may be useful but is perhaps more useful as foundation to the general expansion of (x+a)(x+b).
There are different ways to look at the problem depending on strategy stage. Students could be
encouraged to invent their own enormous square number problems.
Exercise 12
Teaching lessons preceeding this exercise could include:• modelling of the problem
• finding the square root button on the calculator and learning how to use it.
This exercise is an introduction to the complexities fo square root. It is important that the “odd” and
even” number of zeros is introduced. When finding a square root it is useful to pair of the digits and
notice that the answer is often the size of the number of pairs. A 6 digit number has 3 pairs and the
“root” will be in the thousands. A complete set of exercises exploring these properties will be
available eventually on this web site.
Exercise 13
Teaching lessons preceeding this exercise could include:• the use of a variable to represent any or “n”y number.
The main purpose of doing this exercise is to gain a notion of what n+1, 2n etc means. Encourage
dialogue that it all depends on the value of n. It is more important that 2n means double than where
it is on the number line. Iinvite student problems and extend to powers and decimals.
Thinking Inside the Square
Answers to Exercises
Practice excercises with answers (.pdf for whole file, .doc for parts)
Exercise 1
1)
4
(2)
25
(3)
1
4)
16
(5)
81
(6)
100
7)
49
(8)
36
(9)
64
10)
49
(11) 64
(12) 81
13)
36
(14) 25
(15) 4
16)
16
(17) 0
(18) 1
Squares of the numbers 1 to 9 are 1, 4, 9, 16, 25, 36, 49, 64, 81
Noticed? The difference between the squares are odd. The squares are odd, even, odd, even…
The differences are odd because always we add an odd and an even number which is odd and
always two more. The odd/even pattern is the result of squaring an odd or even number which arose
from add + add = even , even + odd = odd and so on. The oddness toggles. Other patterns possible.
Exercise 2
1)
4)
3x10
5x10
(2)
(5)
6x10
9x10
(3)
(6)
7x10
1x10
7)
4x10
(8)
2x10
(9)
8x10
10)
900
(11) 3600
(12) 4900
13)
2500
(14) 8100
(15) 100
16)
1600
(17) 400
(18) 6400
19)
90000
(20) 360000
(21) 490000
22)
250000
(23) 810000
(24) 10000
25)
160000
(26) 40000
(27) 640000
28)
9000000 (6 zeros)
(29) 3600000000 (8 zeros)
(30) 490000000000 (10 zeros) 31)
100000000000000 (14 zeros)
(32) 2500000000000000 (14 zeros)
(33) 490000000000000000 (16 zeros)
34)
1000000000000000000 ( 18 zeros)
(35) 1000000000000000000000000000000000000 (30 zeros)
(36) A googol is 1 followed by 100 zeros so a googol squared is :10000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000 1 followed by 200 zeros!
A googol is 1 follwed by 100 zeros.
The number of zeros or placeholders doubles when a number is squared.
Exercise 3
Choosing 20 + 5 is better than 17+8 because it is easier to deal with.
1)
100 + 50 + 50 + 25
(2)
100 + 20 + 20 + 4
(3)
100 + 70 + 70 + 49
4)
100 + 10 + 10 + 1
(5)
100 + 40 + 40 + 16
(6)
100 + 80 + 80 + 81
7)
100 + 80 + 80 + 64
(8)
100 + 30 + 30 + 36
(9)
100 + 50 + 50 + 25
10)
100 + 30 + 30 + 9
(11) 400 + 20 + 20 + 1
(12) 1600 + 200 + 200 + 25 or another way.
The answers always have 4 parts. Various other notices.
Squares of numbers 11 to 19 are 121, 144, 169, 196, 225, 256, 289, 324, 361
Games can include pickup memory, snakes mystery cards, square dominos, snap.
Exercise 4
Students should notice the L shape gets bigger by two each time and explain why.
Firt 4 odd numbers must add to 16.
First 10 numbers will add to 100.
The sum from 1 of any n odd numbers is nxn or n squared.
Exercise 5
The infinite set may need explaining. The number of grains of sand on all the beaches in the world
is not an infinite set but it is uncountable. The number of odd numbers is infinite because there is
always a bigger one than the biggest one you can name.
Pattern 1
This shows the sum of odd numbers, the sum, and the sum written as a square.
Last row = 1 = 3 + 5 + 7 + 9 + 11 + 13 or 7 terms, 49, 72
Pattern 2
This is as follows
1
3+5
7+9+11
13+15+17+19
1
8
27
64
13
23
33
43
21+23+25+27+29+31
125
53
33+35+37+39+41+43+45
216
63
47+49+51+53+55+57+59+61
343
73
63+65+67+69+71+73+75+77+79
512
83
rd
middle in 3 row is 9 or 3 squared
middle in 11th row is 11 squared or 121
5th row add to 5x5x5 = 125
1th row adds to 11x11x11 = 1331 (nice link here to Pascals triangle)
Other patterns exist! Look at the digit place and how it changes downwards. Odd/even in cubes!
Exercise 6
Other ways are possible to make 4x4. 2 groups of 2x4 or 4 groups of 2x2
Using method 1
1)
81
(2)
81 + 9 + 10 = 100
(3)
4)
121+11+12=144
(5)
144+12+13= 169
(6)
7)
195+14+15=225
(8)
225+15+16=256
(9)
Using method 2
10)
289+2x17 =1=324
(11) 324+2x18=1=361
(12)
13)
400+2x20+1=441
(14) 441+2x21+1=484
(15)
33x33=1024+32+33=1089, using method 1
223x223=49284+2x222+1=49729, using method 2
100+10+11=121
169+13+14=196
256+16+17=289
361+2x19=1=400
484+2x22+1=529
Exercise 7
The first 12 triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78…
T8 = 36 and T10 =55
Recording will be various but should say it always makes a square.
T11+T12 = 12 x 12
289 = 17x17 so T16+T17 will make 289
10000 = 100x100 so T99 + T100 will make 10,000.
A curious extension is “Are there any triangular numbers that are also square?” to which the
answer is of course “no”. A triangular number can be rearranged to make a rectangle but never a
square and the closest rectangle will always have a side that is just one bigger than the other.
Exercise 8
Study, various.
What do you see is various.
The main purpose here is to build the models and feel how they transform from one to the other.
Remaking the T2 with the colours shifted allows the odd numbers to be more clearly seen. The
bottom diagram shows this.
Exercise 9
If we subtract an odd then we must subtract an even and if we subtract an even then we must
subtract an odd because the shape is always reduced by 1.
1)
900
(2)
900-30-29=841
(3)
625
4)
625-25-24=576
(5)
4900
(6)
4900-70-69=4761
7)
5625
(8)
5625-75-74 = 5476
(9)
10,000
10)
10000-100-99=9801
(11) 62500
(12) 62500-250-249=62001
49x49=2500-50-49 = 2401
999x999=1000000-1000-999=998001
36x36=1369-37-36=1296
Exercise 10
1)
81
(2)
81+2x9x2+2x2=121
3)
484
4)
484+2x22x2+2x2=576
5)
1764
(6)
1764+2x42x2+2x2=1936
7)
2601
(8)
2601 +2x51x2 + 2x2 =2809
9)
7921
10)
7921+2x89x2+2x2= 8281
11)
14400
(12) 14400+2x120x2+2x2=14884
34x34=1024 + 2x32x2+4 = 1156
224x224 = 49284 + double double 222 + 4 = 50176
If the increase is n then the next square is 2xnxthe original number plus nxn
Since 20 x 20 = 400, so 27x27 = 400 + 2x7x20+7x7= 729
Exercise 11
1)
2x300+25=625
(2)
3x400+25=1225
4)
7x800+25=5625
(5)
1x200+25=1225
7)
10x1100+25=11025
(8)
11x1200+25=13225
10)
15x1600+25=24025
(11) 18x1900+25=34225
Why does it work?
Using the array model we see the the calculation is;
100xn2 + 2x50xn +25= 100(n2 + n) +25 = = 100x(n(n+1) +25
where the n(n+1) is the product of two consecutive numbers
and the 100x pushes this into the hundreds column
which leaves the tens and units column clear for the 25.
A neat trick…can you find another?
(3)
(6)
(9)
(12)
5x600+25=3025
8x900+25=7225
12x1300+25=15625
19x2000+25= 38025
(3)
(6)
(9)
(12)
a bit more than 5
a bit more than 8
nothing like 50
nothing like 80
Exercise 12
1)
4)
7)
10)
5
8
nothing like 50
nothing like 80
(2)
(5)
(8)
(11)
a bit less than 5
a bit less than 8
nothing like 50
nothing like 80
Exercise 13
Various answers. As long as the student can explain why the n is where it is placed and the
explanation fits then the answer is OK.
Eg n can be to the left of 0 because n can be any where. 2n must be twice as far to the left if that is
the case.
The notion of what the expression means is the important part. In this example “twice as far”.