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SMILE CENTRE MIDDLE ROW SCHOOL
KENSAL ROADLONDON W10 5DB
Tel. 01-960 7330
You will need Worksheet 1902a
1. Cut out all the triangles on worksheet 1902a and sort them into 2 sets of similar triangles by comparing angles.
2. Measure the sides of each triangle and complete the tables on your worksheet.
3. You should have found that the ratio Short - Middle is approximately the same for all the triangles
^ in each set.
Is this true for the other 2 ratios?
4. Draw a new set of similar triangles and check that the ratios of their sides are approximately the same.
Smile Worksheet 1902a
Length of sides Ratio of sidesSetl
Short Middle Long Sf-M S^L M-L
Set 2
Short Middle Long S-J-M S-L M-L
Your own set
Short Middle Long S-nM S-L M-L
© RBKC SMILE 2001
Smile Worksheet 1904
Find the operationIn this operation table
a * b means treble a then add b
so a * b = 3a + b
1. Complete the table.
What number patterns do you notice?
2. Find the operation in these tables.
*0
12
3
0
0
0
0
0
1
0
2
4
6
2
0
4
b
12
3
0
6
12
Id
a* b = a * b =
a
*
0
1
2
3
4
5
0
6
1
13
2
2
3
6
12
4
19
5
20
*
0
1
2
3
0
0
1
4
9
1
-1
0
3
5
2
-2
-1
2
7
3
-3
-2
1
6
a * b =
*
0
1
2
3
0
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
Use the sheet below to make a puzzle for a friend.
Find the operation
Designed by
for
Solution
a *b =
*
0
1
2
3
4
0 1 2 3 4
© RBKC SMILE 2001
Smile Worksheet 1907
Work with someone else.
a. Discuss how long it takes to do each of these.Then time yourself.
How close were you?You will need
a clock or a watch.
"I J Count up to 100 2 j Count down from 100
1, 2, 3, 4, ... 100,99,98...
S Actual time
3 ) Write your name 4 / Do 12 x 4 without a calculator
;J Take the register
I Actual time
Make one big cube using 27 smaller cubes
b. Choose 3 of these and then do some of your own.
D^/ Boiling a kettle '\
5 j Getting home from school
2 } Eating a packet of crisps
© RBKC SMILE 2001
Smile Worksheet 1911
Dissection Pairs
Shape 1,.. I can be cut and rearranged to give ... Shape 10.
Find 5 other pairs of shapes.
2) 3)
4) • • • • 5) ' 6)
7) 8) ' ' ' ' 9)
10) ' ' ' ' 11) 12) '
Check your answers by comparing areas.© RBKC SMILE 2001
Smile 1912
Painted TyresWhile riding my bicycle along a path I went through a small patch of paint.
A little later I looked back at the paint marks left by my tyres.
What did I see?
There are some hints on the back if you need them.
HintsIt may help to make a scale drawing of any patterns formed showing the original patch of paint.
* What happens if you change the distance between the wheels ?
* What happens if the diameter of the wheels is different ?
* What happens if you turn a corner ?
* Do both the wheels travel the same distance ?
Smile 1913
8
8
8
bo
This is part of an addition table in Bengali number script.
Can you extend it ?
Smile Worksheet 1914
•Adding Counters
You will need counters.
Add one more counter to make 6 lines of 3 counters.
Add two more counters to make 3 lines of 3 counters.
Add two more counters to make 5 lines of 4 counters.
turn over
Add four more counters to make 8 lines of 3 counters.
© RBKC SMILE 2001
Smile 1916
Domino TrickAsk a friend to choose any domino.
Let's try this one...
Ask her to do the following calculations.
# Multiply either of the numbers by 5
# Add 8
Multiply by 2
Add the other number from the domino
Ask her for the answer. Subtract 16 from it.
The two digits left are the numbers on the domino.
Will it work with any domino ?
Can you prove it ?
Can you find another set of rules with the same effect?
cSmile 1917
^--
!vV
^i|j
i
1i
i
t
-/
\
p\
i
-* \"J\
-/•w
ov>tfV^
ti
iii
**iiii
ii
,\i>^\N
&
ii|i
i~~ri^ ^'
1J5
Ii- —
iji
•- H
\3
>/it
V
rirx"i
^
f*\£\c
X
0h
^r
*
\
s
*1»—
11j
^
\f^*
x
w0,<^
111
1
11
,^' 1
1
1
-r;3^
00--*•
i
iS*
1 1
I
ietf^"
Ii1ii
i
>X* iiii
i
i iiii1
(*<
v\o^<
-f-
jS*
\
%
\c
x
G<•a/^
^^
^ \c
^
X•-•
^
^-»
s-*/
f
*f
0p
1 n
i
\^\
-14-! I
1
1
1
1
———— 1
-!iiii
i^F^1 'i •i • .i : -
ii ;iiii i
1 1i ••ii .1 11 1 *ii i
Gradient measures how steep aslope is.
Gradient = he'9ht base
So for this triangle
8cm
gradient = 2.7 -=- 8
= 0.34
Check that all of these have a gradient of approximately 0.58
1. By drawing right angled triangles find the gradients for other angles. Copy and complete the table.
Angle Gradient
0°
10°
20°
30°
40°
50°
60°
70°
80°
0
0.58
2. Draw a triangle to show a gradient of exactly 1.
What angle has this gradient ?
Turn over
The tangent buttona scientific calculator will giveyou a gradient for any angle.
3. Use the tan button to checkyour results in the table.
Explore the gradient for angles between 80° and 90°-
4. What happens to the gradient for angles very near 90° ?Hint:Look at angles between 89° and 90°
Investigate the tangent of angles greater than 90°
Smile 1918
The Coin ProblemPut down 5 coins — all showing TAILS.
Can you make all 5 coins show HEADS by turning them over two at a time ?
Try starting with 6 coins, 7 coins., .
Investigate turning over three at a time, four at a time
How many centimetre squares?Smile Worksheet 1919
&
© RBKC SMILE 2001
Trig LinesYou will need graph paper and a scientific calculator.
Smile 1921
.9
---0.8
0.7
-- 0.6
0.5
*'-
>.4
---0.3
-- 0.2
opDasiI _.i. ll__ __I __ ___ L__ ___
side
^35° ad acent side^
rt w= a aThe red line (radius) is 1 unit long. )lt rotates anti-clockwise. It has stopped at 35°.
1 . Look at the right-angled triangle formed bydrawing a vertical line opposite to the 35° angle. The length of the side opposite to the 35° angle is 0.57.
a. How long is the side adjacent to the 35° angle?
b. On your graph paper draw diagrams with the red line stopped at 30°
45° 60° 80°
Measure the opposite and adjacent sides for each angle.
opposite side
c. How does the length of the opposite side change as the angle gets larger?
How does the length of the adjacent side change as the angle gets larger?
When the radius is 1 unit long the opposite and adjacent sides can be found using the [siiT] and [cos] buttons on a scientific calculator.
Press I 3 I |s answer for 1 a.
cos) to check your
Use the sine and cosine button to check that your answers to 1b. are approximately correct.
2. Find the length of the opposite and adjacent sides of these triangles. (They are not drawn to scale.)
^25°
c.
e.15 C
d..40°
Turn over
The small triangle has been enlarged by Scale Factor 3.
3. What are the lengths of the sides on the larger triangle?
4. Find the lengths of the sides on thesetriangles.(They are not drawn to scale.)
a.
c.35 l
63
20
Matrices and AreaSmile 1922
5 -
4 -
3 _
2 -
1 -
(2, 3) (4) 3)
' (2,1)
——— i ——— i
iiiiiiiiiiiiliiiiiiiiiiiii
1 —————— 1 ——————— L
(4,1)
——— i —
123
1. Transform this square using the matrix2 o o 3
4 4 2
l 1 3 3
* 8
s •
What is the area of the original square?
What is the area of the new shape?
What is the ratio New Area: Original Area?
Turn over
2. Transform the same square using the following matrices:
;i 0 ) 3
3 1
1 2
siSilS^^
© 1991 RBKC SMILE
You will need a set of 12 wooden pentominoes.
Pentomino PuzzlesSmile 1927 (
This 4x5 rectangle has been made using four of the pentominoes.
Use 2cm squared paper to show how you solved the three puzzles inside.
Make a rectangle with these 4 pentominoes.
Make a rectangle with these 6 pentominoes.
Turn
Make a rectangle using these 7 pentominoes.
© 1991 RBKC SMILE
r~
You will need a set of 12 wooden pentominoes. Smile 1928
four pentominoes
An enlarged copy of the n | pentomino can be
made using four other pentominoes.
1 . Make an enlarged copy of the | ___ | pentomino
using four other pentominoes.
. Do the same for <v ^ ^ } and
Draw your results on 2cm squared paper.
3. a) How do the sides of the enlargement compare with the sides of the single pentomino?
b) How does the area of the enlargement compare with the area of the single pentomino?
Turn over
One other pentomino can be enlarged in this way.
Can you find it?
© 1991 RBKC SMILE
Smile 1929'
You will need a set of 12 wooden pentominoes and 2cm squared paper.
Select one pentomino.
Use nine of the other pentominoes l^make an enlargement of it. Draw your result,
Try some more.
© 1991 RBKC SMILE
How many different numbers ? How many different scripts ? Which scripts ?
(0.p
O CO
O •H
CO O)
EO)
If you want more information look at Worksheet 1931 a
Smile Worksheet 1931 a
Which Scripts?There are 6 numbers written in 5 different scripts.
Can you find them?
Write 51 in each script.
© RBKC SMILE 2001
Smile Worksheet 1931 a
Which Scripts?
There are 6 numbers written in 5 different scripts.Can you find them ?
Write 51 in each script.
Smile 1934
a mWfou may need tracing paper.
A translation moves every part of a shape the same direction the same distance
This shape ... is moved by this translation to this new position.
Show these translations on squared paper.
1. 2.
3. 4.
Combined i ranslafions
This shape ... ... is moved bythese two translations ... to this position,
5. Show the following translation on squared paper.
Find a single translation to replace these two translations.
Turn over
Find a single translation to replace those in each of 6 and 7.
6. 7.
© 1991 RBKC SMILE
You will need filter papers.
Angles in semi-circles
Smile 1935
Fold a filter paper in half. Draw different triangles using the fold as the base.
Investigate the angles of your triangles,
Does the size of the circle matter?Turn over
What happens to the angles if P is inside the circle ?
... outside the circle?
Smile 1937
Panjabi Numbers
tf ++ 3 =
x tf + 8 =
x tf +
+
+ 'J
t
Copy and complete this number pattern in Panjabi number script.
1991 RBKC SMILE
Olympic Rlfedals Smile 1938o~
f LEADING 15 MEDAL WINNERS IN THE LAST FIVE GAMES II
| MUNICH 1972i °Soviet Union........ «»1 United States. ,„..
• Italy......................
• Finland..... ....... ......• Cuba. _ .
•M
, ?011
.... 13
...... 8
...... 7
...... 6
...... 6
...... 5
_ 3.. 3
S27 31 23 11 8 7 5
13 10 3 6 5 6 1 1
B 2230 23 16 8 2 9
16 5
10 6 9 7 4 4
Tot99 94 66 40 29 17 21 35 21 18 16 18 16 8 8
MONTREAL 19Q
Italy............... .Franco ... _ .__.....
.... 4034
.... 109
...... 7
...... 6
...... 6
...... 4
...... 2
76s
4125 35 12
6 6 9 4 9 5 2 1 5 7 3
B35 25 25 17 10 13 7 3
14 13 0 0 5 4 4
Tot125 90 94 39 25 26 22 13 27 22
6 5
13 13 9
MOSCOW 198CG
Czechoslovakia .....
4788fl
F,fi5
.... 333
....2
.». 2
.... ?
}
S69 37 16 7 3
10 6 5 7
14 3 1 3 3 2
B46 42 17
5 4
15 13 3 9
15 6 4 9 4 2
Tot195 126
4120 15 32 25 14 21 32 12 8
14 9 9
LOS ANGELESG
United States ..„—.— 83
Italy............ . ....
Great Britain.....
Australia ........... Finland.............
....... 20
....... 17
....... 15
....... 14
....... 10
....... 10
........ 8
......... 7
......... 6
......... 5
......... 5
......... 5
......... 4 "........ 4
_«_^ri
1984S B
61 31 16 17 19 23 8 9 6 12
18 16 8 14 1 2 4 7 6 7
11 21 7 16 2 6 8 12 2 6
^^•^M
Tot175 53 59 32 32 44 32 11 18 19 37 28 13 24 12
fc
SEOUL 1988G
Soviet Union............. 55East Germany.... United States .....Sooth Korea.......West Germany.... Hungary...............Bulgaria. ........ ......Romania... ...........F'ance .................Italy......................China...................Great Britain........Kenya .......... ........Japan . .. ....Australia ............
..... 37
..... 36
..... 12
..... 11 .... 11
..... 10
....... 7...... 6
....... 6...... 5...... 5...... 5...... 4...... 3
S31 35 31 10 14 6
12 11 4 4
11 10 2 3 6
^^^m
B46 30 27 11 15 6
13 6 6 4
12 9 2 7 5
^H
Tot 132 102 94 33 40 23 35 24 16 14 28 24 9
14 14
IB—-
Source: The Sunday Times, October 1988
This graph shows the mecwlfe won by the Soviet Union.
KeyBronze
Silver
Gold
CO T3 0)
D Z
200
150
100
50
1972 1976 1980 1984 1988Years when Olympics were held.
1. Which countries medals are shown by these graphs?
1972 1976 1980 1984 1988Years when Olympics were held.
TO 130)
3z
40
30
20
10
1972 1976 1980 1984 1988Years when Olympics were held.
2. Pick another country Ad draw your own graph. Ask someone to find out which country.
You will need a scientific calculator and graph paper. Smile 1939
and fcoiT graphs
Draw a table showing the values of sine and cosine for angles between 0° and 360°.
Turn over
Plot three graphs to show
• sine against angle• cosine against angle• sine against cosine.
Describe your three graphs.
What happens for angles greater than 360°? What happens for negative angles ?
Dividing Investigation Smile 1940f
Which other numbers give answers ending in ,25 when you divide them by 4?
Try dividing by some other numbers: 5, or 8, or 3 , , ,
Write about what you have found.
Smile 1941
Differences™The mapping n —>3n2 + n creates the sequence 4, 14, 30, 52, 80, 114...
Taking the differences betweenthe terms of the sequence 4 14 30 52 80 114
gives 10 16 22 28 34
and again 6666
wThis gives a constant difference of 6 after two rows.
Investigate for other mappings. You may wish to use a spreadsheet.
© 1991 RBKC SMILE
Smile Worksheet 1942
Growing Patterns
turn over
• Now try with your own shapes.
© RBKC SMILE 2001
Smile Worksheet 1945
Square ^Diagonals
This square has been filled in using the rule
"Add 1" acrossand
"Add 1" down.
+1
1
2
3
4
5
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8>
5
6
7
d
9
Complete this square in the same way.
1 +
You should find numbers on thi diagonal.
What do you n
r 1
th
oti
i ^11
A-
6
ese
ce about t
9
&
12
w
'
;;«;; gf
>
11 W-hem?
4
4What do you notice about the numbers on the other diagonal?
Complete this square using
"Add 2" acrossand
"Add 2" down.
+2
0 4
14
What do you notice about the numbers on the diagonals?
turn over
This sqgare has two different rules.
"Add 2" across and
"Add 3" down.k. .0
r 3
^ T *-
0
3
12
15
d
What do you notice about the numbers on the diagonals?
If you enjoyed doing this you might like to investigate what happens for different rules and different squares.
© RBKC SMILE 2001
Smile 1946
© 1991 RBKC SMILE
A matrix can describe a framework by listing the connecting lines.
Smile 1947$
Which matrices opposite describe the frameworks below?
1.
4.
a. b. c. d.
A
FD
E
A
01
1
1
0
B
10
1
1
1
C
11
0
11
D
11
1
0
1
E
01
1
1
0
e.
A
B
C
D
E
F
G
A
0
1
1
1
1
1
1
B
1
0
1
0
0
0
1
C
1
1
0
1
0
0
0
D
1
0
1
0
1
0
0
E
1
0
0
1
0
10
F
1
0
0
0
1
0
1
G
1
1
0
0
0
10
A
B
C
DlE
F
A
0
1
1
1
1
0
B
1
0
1
0
1
1
C
1
1
0
1
0
1
D
1
0
1
0
1
1
E
1
1
0
1
0
1
F
0
1
1
1
1
0
f.
A
B
C
D
E
F
A
0
1
0
0
1
1
B
1
0
1
1
0
0
C
0
1
0
1
0
1
D
0
1
1
0
1
0
E
1
0
0
1
0
1
F
1
0
10
10
A
B
C
D
E
F
A
0
1
1
0
1
0
B
1
0
1
0
0
1
C
1
1
0
1
0
0
D
0
0
1
0
1
1
E
1
0
0
1
0
1
F
0
1
0
1
1
0
g.
A
B
C
D
E
A
0
1
1
1
1
B
1
0
1
0
1
C
1
1
0
10
D
1
0
1
0
1
E
1
1
0
1
0
A
B
C
D
E
F
G
H
A
0
1
0
0
0
110
B
1
0
1
0
0
0
0
1
c0
10
10
0
10
D
0
0
10
10
0
1
E
0
0
0
10
110
F
1
0
0
0
10
0
1
G
1
0
10
10
0
0
H
0
1
0
1
0
1
0
0
Can you draw a network for the remaining matrix?
Smile 1948
y = ax
Draw graphs for different values of a.
• How does the value of a change the graph?
• What happens if a is negative?
Try using a graphic calculator or a graph plotting program such as QUAD.
© 1991 RBKC SMILE
Smile 1949
Each player needs a counter.or centicube.
You will need a dice marked
N EW
NESW
NWSE
NW NE
SWSE
Take turns to throw the dice.
Choose one of the directions shown. Move one square in that direction.
The winner is the first person to reach a shaded square.
North (N)North ^-^^lifcs^ North West East (NE)
West (W) East (E)
South West (SW)
South East (SE)
South (S)
Start
Smile 1905
SortingTriangles
Sort the triangles in this pack into sets of similar triangles.
What is special about similar triangles?
© RBKC SMILE 2001
