Welcome!
This is the fourth in a series of teaching aids designed by teachers for teachers at level 7/8.
The worksheets are designed to support the delivery of the National Curriculum in a variety of
teaching and learning styles. They are not designed to take the pedagogy away from the teacher.
The worksheets are centred around the shown level, but spiral from the level below to the level
above. Consult the National Numeracy Strategy for definitive National Curriculum levels.
They can be used by parents with the support of the on-line help facility at www.10ticks.co.uk.
Contents and Teacher Notes.
Pages 3/4.
Pages 5/6.
Pages 7/8.
Pages 9/10.
Pages 11/12.
Pages 13/14.
Pages 15/16.
Pages 17/18.
Pages 19/20.
Level 7/8 Pack 4. Page 1.
Graphical Intersection.
The worksheet starts solving simultaneous equations through graphical
intersection. Pupils can see why there is an x answer and a y answer. This point
can be reinforced when solving them algebraically at a later date. The main
focus of the sheet is solving quadratics through graphical intersection.
Simultaneous Linear Equations (Elimination).
Solving simultaneous equations through elimination. The sections are
progressively structured through the required skills.
Simultaneous Linear Equations (Other Methods).
Rearranging equations ready for elimination. Substituting one equation into
another and some basic trial and improvement solutions.
Simultaneous Linear Equations (Worded Questions).
The first section concentrates on the skill of putting a worded question into
two simultaneous equations. The second section pulls all the skills of solving
simultaneous equations together.
Puzzling Algebra (Simultaneous Equations).
Puzzles where objects represent numbers. To solve the puzzle pupils need to be
able to form and solve simultaneous equations.
Inequalities.
Traditional exercises solving inequalities.
Inequalities (Worded Questions).
As with Simultaneous Equations (Worded Questions), the essential skill is to
be able to turn the words into an inequality ready for solving. The questions
progressively increase in complexity.
Graphical Inequalities 1 (without equations).
Suitable for pupils moving on to Higher Level GCSE or for the more able
Intermediate pupils. Pupils have to identify the equation of the boundary line
then give an inequality for the shaded area. Straight line graphs need to be
revisited before attempting the sheet.
Graphical Inequalities 2 (without equations).
The complexity of the inequality needed to describe the shaded area increases.
Pupils now have to draw required inequalities.
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Pages 21/22. Graphical Inequalities 1 (with equations).
This is the previous sheets repeated, except with the boundary equations
marked. This eliminates the skill of finding the equation of a line and focuses
on inequalities only. Far better for weaker Intermediate GCSE candidates.
Page 23.
Graphical Inequalities 2 (with equations).
As above, Page 20 is the reverse of this sheet.
Pages 24-30. Inequality Snap Cards.
The algebra, number line and graphical representations are brought together.
Page 24 and 25 are the opposite algebraic ways of writing the equation. It is
interesting for pupils to compare the number line to the graphs. One exercise
may be to cut out the cards and stick them into books in sets. Another is snap.
Page 31.
Translations.
You should be able to fit 4 questions on one side of A4. All you need to know
about translations.
Page 32.
Reflections.
Reflection questions.
Page 33.
Rotations.
Rotation questions.
Pages 34/35. Enlargements.
Enlargements covering fractional and whole number enlargements.
Page 36.
Constructing Enlargements.
Practical exercise constructing enlargements using the ‘ray’ method. The centre
of enlargement and scale factor are given, pupils have to construct the
enlargement.
Pages 37/38. Transformations.
An exercise that covers all the transformations needed at this level.
Pages 39/40. Combining Transformations.
The effect two transformations can have upon a given shape. Each
transformation is defined by a letter and the notation SV(A) means the shape A
is transformed by V then by the transformation S.
Pages 41/42. Transformation Investigations.
Some investigations pupils can attempt that involve transformations.
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are trademarks of Fisher Educational Ltd in the UK and/or other countries.
Details of copyright ownership in the clip art used in these worksheets:
Copyright in the clip art used entirely in this pack is owned by Nova Development Corporation, California, USA.
Level 7/8 Pack 4. Page 2.
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Graphical Intersection.
Simultaneous Linear Equations.
Plot each set of linear equations using the "x = 0, y = 0" method, and solve them.
1).
2x + 3y = 24
x + 2y = 14
2).
x + y = 8
x + 2y = 14
3).
3x + 2y = 18
3x + 4y = 24
4).
2x + y = 11
3x - 2y = 6
5).
3x + 2y = 16
2x + y = 9
6).
2x + 5y = 17
5x + y = 31
7).
x + y = 5
3x + y = 9
8).
x + y = 6
2x + y = 10
9).
x - y = 2
2x - y = 6
10). y - 2x = 5
y - 3x = 2
11). 2x - y = 10
x + 3y = 12
12). 4x - y = 1
x + 2y = 16
13). 3x + y = 10
4x + y = 13
14). 2x + y = 9
3x + 2y = 14
15). 3x + y = 11
x + 2y = 7
16). 3x - y = 8
x - 2y = 1
17). y - x = 7
y - 4x = 1
18). y - 3x = 1
2y - x = 12
19). 3x + 40y = 26
2x + 10y = 9
20). 2x + 3y = 9
4x - 2y = 2
21). 2x + 4y = 16
5x + 2y = 20
22). 2y - 3x = 2
4y + 3x = 13
23). x + 4y = 11
5x + 6y = 13
24). 8y + x = 6
3y - 2x = 7
25). 2x - 3y = 15
x - 4y = 10
26). 2x - 4y = 10
3x + 2y = 7
27). 2x - 3y = 12
4y + 5x = 7
28). 2y - 3x = 4
x - 4y = 2
29). 2x - y = 4
-x - y = 7
30). 2x + 5y = -10
4x + 3y = -13
Quadratic Equations.
1).
Plot the equation y = x2 - 3x
a).
2).
x2 - 2x = 0
x2 - x = 0
b).
x2 - 3x = 2
c).
x2 - 3x = -1.
for -3 ≤ x ≤ 5. Use this graph to solve
b).
Plot the equation y = x2 - x
a).
4).
b).
Plot the equation y = x2 - 2x
a).
3).
x2 - 3x = 0
for -2 ≤ x ≤ 5. Use this graph to solve
x2 - 2x = 10
c).
x2 - 2x = 5.
for -3 ≤ x ≤ 4. Use this graph to solve
x2 - x = 2
c).
x2 - x = x
d).
x2 - x = x + 1.
d).
2 - x2 = -x.
Plot the equation y = 2 - x2 for -3 ≤ x ≤ 3. Use this graph to solve
a).
Level 7/8 Pack 4. Page 3.
2 - x2 = 0
b).
2 - x2 = -4
c).
2 - x2 = x
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5).
Plot the equation y = x2 - 5x + 4
a).
d).
6).
b).
e).
x2 - 7x + 10 = 0
x2 - 7x + 10 = x - 3
b).
e).
Plot the equation y = x2 - 2x - 8
a).
d).
9).
x2 - 6x + 9 = 0
x2 - 6x + 9 = x - 2
Plot the equation y = x2 - 7x + 10
a).
d).
8).
b).
e).
Plot the equation y = x2 - 6x + 9
a).
d).
7).
x2 - 5x + 4 = 0
x2 - 5x + 4 = x + 1
x2 - 2x - 8 = 0
x2 - 2x - 8 = -2x - 3
b).
e).
Plot the equation y = 2x2 - 9x + 4
a).
d).
2x2 - 9x + 4 = 0
2x2 - 9x + 4 = -x + 3
b).
e).
10). Plot the equation y = x2 + 2x - 3
a).
d).
x2 + 2x - 3 = 0
x2 - 2x - 3 = -0.5x
b).
e).
for -1 ≤ x ≤ 6. Use this graph to solve
x2 - 5x + 4 = 4
x2 - 5x + 3 = 0
c).
f).
x2 - 5x + 4 = x
x2 - 5x + 6 = 0.
for 0 ≤ x ≤ 6. Use this graph to solve
x2 - 6x + 9 = 6
x2 - 6x + 5 = 0
c).
f).
x2 - 6x + 9 = x
x2 - 6x + 7 = 0.
for 0 ≤ x ≤ 7. Use this graph to solve
x2 - 7x + 10 = 5
x2 - 7x + 7 = 0
c).
f).
x2 - 7x + 10 = x
x2 - 7x + 12 = 0.
for -3 ≤ x ≤ 5. Use this graph to solve
x2 - 2x - 8 = -6
x2 - 2x - 9 = 0
c).
f).
x2 - 2x - 8 = 0.5x
x2 - 2x = 0.
for -1 ≤ x ≤ 5. Use this graph to solve
2x2 - 9x + 4 = -5
2x2 - 9x + 2 = 0
c).
f).
2x2 - 9x + 4 = -x
2x2 - 7x + 5 = 0.
for -5 ≤ x ≤ 3. Use this graph to solve
x2 + 2x - 3 = 4
x2 + 2x - 4 = 0
c).
f).
x2 + 2x - 3 = x
x2 + x - 3 = 0.
11). Plot the equation y = x2 - 2x + 4 for -3 ≤ x ≤ 5. Use this graph to solve
a).
d).
x2 - 2x + 4 = 6
x2 - 2x + 4 = -x + 4
b).
e).
x2 - 2x + 4 = 10
x2 - 2x + 1 = 0
c).
f).
x2 - 2x + 4 = 3x
x2 - 4 = 0.
12). Plot the equation y = 2 + x - x2 for -3 ≤ x ≤ 4. Use this graph to solve
a).
d).
2 + x - x2 = 0
2 + x - x2 = -x + 2
b).
e).
2 + x - x2 = -4
8 + x - x2 = 0
c).
f).
2 + x - x2 = x
-x - x2 = 0.
13). Plot the equation y = x2 + 3x - 7 for -5 ≤ x ≤ 2. Use this graph to solve
a).
d).
x2 + 3x - 7 = 0
x2 + 3x - 7 = -x - 2
b).
e).
x2 + 3x - 7 = -4
x2 + 3x = 0
c).
f).
x2 + 3x - 7 = x
x2 + x - 2 = 0.
14). Plot the equation y = x2 - 5x + 2 for -1 ≤ x ≤ 6. Use this graph to solve
a).
d).
x2 - 5x + 2 = 0
x2 - 5x + 2 = x - 4
b).
e).
x2 - 5x + 2 = -3
x2 - 5x = 0
c).
f).
x2 - 5x + 2 = -x
x2 - 3x = 0.
15). Plot the equation y = -x2 + 6x - 5 for 0 ≤ x ≤ 6. Use this graph to solve
a).
d).
Level 7/8 Pack 4. Page 4.
-x2 + 6x - 5 = 0
-x2 + 6x - 5 = -x + 5
b).
e).
-x2 + 6x - 5 = 3.5
-x2 + 6x - 7 = 0
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c).
f).
-x2 + 6x - 5 = x
-x2 + 5x - 4 = 0.
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Simultaneous Linear Equations (Elimination).
Solve the following simultaneous linear equations.
A.
1). 3x + y = 7
2). 2x + y = 21
3).
2x + y = 6
x-y=6
x + 5y = 21
x + 2y = 12
7).
5x + y = 28
x-y=2
4).
3x + y = 25
x-y=3
x + 6y = 20
x + 3y = 11
8).
x + 5y = 11
x + 4y = 10
5).
x + 9y = 13
x + 3y = 7
6).
9).
x+y=7
2x - y = 2
10). x + y = 11
5x - y = 25
11). x + 8y = 26
x+y=5
12). x + 5y = 25
x + 3y = 17
13). 4x + y = 19
x + y = 10
14). x + 5y = 15
x+y=7
15). 4x + y = 23
3x + y = 18
16). x + 6y = 17
x + 2y = 9
17). g + h = 6
-g + 3h = 10
18). a + 4b = 18
-a + 7b = 15
19). 2f + g = 13
6f - g = 3
20). p + 6q = 16
p + 2q = 8
1).
3x + 4y = 20
2x + y = 10
2).
5x + 2y = 16
4x + y = 11
3).
3x + 2y = 16
2x + y = 9
4).
5x + 4y = 23
3x + y = 11
5).
3x + y = 13
5x - 2y = 7
6).
2x + y = 11
3x - 2y = 6
7).
x+y=7
11x - 4y = 2
8).
6x + y = 20
4x - 3y = 6
9).
5x + 3y = 14
x + 2y = 7
10). 3x + 4y = 24
x + 5y = 19
11). 2x + 5y = 16
x + 3y = 9
12). 3x + 2y = 17
x + 3y = 15
13). x + 4y = 6
-4x + 9y = 1
14). x + 3y = 10
-2x + 5y = 2
15). x + 5y = 16
-2x + 3y = 7
16). x + 4y = 22
-4x + 5y = 17
17). 3x + 2y = 13
2x - y = 4
18). 4x + 3y = 17
3x - y = 3
19). 3x + 2y = 16
x-y=2
20). 10x + 3y = 19
5x - y = 2
21). 2x + 5y = 31
x + 6y = 33
22). 2x + 3y = 16
3x + y = 17
23). 3x + 4y = 25
x + 2y = 11
24). 6x + 5y = 23
2x + y = 7
25). 5t + 2u = 33
-t + 5u = 15
26). 2p + 11q = 34
-p + 7q = 8
27). 3r + 4t = 10
2r - t = 3
28). 6n + 5m = 32
3n + m = 10
1).
2x + 3y = 9
5x + 4y = 19
2).
2x + 5y = 16
3x + 4y = 17
3).
5x + 2y = 31
2x + 3y = 19
4).
3x + 5y = 25
2x + 3y = 16
5).
4x + 3y = 15
5x - 4y = 11
6).
3x + 2y = 14
13x - 5y = 6
7).
3x + 2y = 13
5x - 3y = 9
8).
2x + 5y = 16
3x - 2y = 5
9).
3x + 4y = 14
-2x + 5y = 6
10). 5x + 6y = 17
-4x + 5y = 6
B.
C.
Level 7/8 Pack 4. Page 5.
11). 2x + 5y = 13
-3x + 16y = 4
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12). 2x + 7y = 25
-5x + 8y = 14
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13). 2y + 5x = 31
7y - 2x = 11
14). 2y + 3x = 18
9y - 4x = 11
15). 5y + 3x = 11
7y - 2x = 3
16). 2y + 3x = 13
7y - 2x = 8
17). 3x + 4y = 25
5x + 3y = 27
18). 3y + 4x = 16
4y + 5x = 21
19). 2x + 3y = 11
16x - 5y = 1
20). 3y + 2x = 15
7y - 5x = 6
21). 9x + 3y = 30
5x - 2y = 2
22). 3x + 5y = 22
2x + 7y = 22
23). 4x + 3y = 22
3x - 4y = 4
24). 3x + 5y = 19
2x + 7y = 20
25). 4d + 3e = 32
5d + 2e = 33
26). 2u + 7v = 17
3u + 5v = 20
27). 2a + 3b = 16
3a + 4b = 23
28). 5q + 7r = 24
2q + 3r = 10
1).
4x + y = 15
3x + y = 8
2).
9x + y = 14
7x + y = 8
3).
3x + y = 24
x+y=6
4).
6x + y = -46
x + y = -6
5).
3x + y = -7
8x + y = -7
6).
x + y = -15
2x + y = -8
7).
x + 6y = 21
x + 9y = 9
8).
x + 5y = 2
x + 3y = -4
9).
x + y = 51
x + 8y = 9
10). x + 9y = 53
x + 3y = 5
11). x + 9y = 28
x + 5y = 0
12). x + y = -10
x + 3y = 0
13). -2x + y = -15
8x + y = -5
14). -3x + y = 30
x + y = 10
15). -9x + y = -39
2x + y = -6
16). -x + y = 2
4x + y = -3
17). 6c + d = -9
c + d = -4
18). 9g + h = -12
-g + h = -2
19). 8a + b = -13
a + b = -6
20). n - 7m = 2
n - 8m = 3
1).
x - 7y = 23
3x + 9y = 9
2).
x + 8y = 26
5x + 9y = 6
3).
9x + 9y = 36
5x + 8y = 11
4).
7x - y = 1
8x + 5y = -5
5).
2x + 6y = -4
8x + 5y = 3
6).
8x + 3y = 23
6x + 5y = 9
7).
y - 2x = -11
6y + 2x = -10
8).
9x - 6y = -66
7x - 7y = 7
9).
2x - 5y = 7
-2x + 8y = -10
10). -5x + 6y = 44
9x + 7y = -8
11). 6x - 8y = 108
7x + 2y = -10
12). 8x + 3y = -17
2x + y = -7
13). -9x + 3y = 3
x + y = -3
14). x + 2y = -13
-x + 3y = -2
15). 5x + 9y = -37
-8x + 4y = 4
16). 9x + 5y = -118
-3x + 2y = -1
17). 2p - 8q = -2
5p - 6q = -5
18). 2j - 2k = 8
7j - 5k = 12
19). 8w + 5v = -15
9w + 2v = -6
20). 5a + b = 38
2a + 4b = 8
1).
4x - 5y = 5
2x - 3y = 2
2).
6x - 2y = 9
3x + 4y = 12
3).
6x - 5y = -7
3x + 4y = 16
4).
3y - 4x = 1
6y - 6x = 5
5).
4x + 3y = 4
2x - 5y = 15
6).
4x + 2y = -8
6x - 2y = -27
7).
8m + 4n = 7
6m - 8n = 41
8).
5a + 10b = 28
15a - 20b = -121
D.
E.
F.
Level 7/8 Pack 4. Page 6.
Licensed to Dr Challoner's Grammar School
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Simultaneous Linear Equations (Other Methods).
Elimination after Rearranging.
Rearrange these equations then solve using elimination.
1).
3x + y = 5
4x = 2 + y
2).
8x = 8 - 3y
5x = 5 + 3y
3).
3y = 4x + 5
4x + 5y = 19
4).
2x = y + 1
5x + y = 13
5).
3x = 1 + y
5x - y = 3
6).
x = 2y + 7
x - 6y = 3
7).
5x = 17 - 4y
2x + 4y = 8
8).
9x = -3y
5x + 3y + 4 = 0
9).
9x + 3y = 19
6x = 11 + 3y
10). 2x + y + 2 = 0
7x + y - 3 = 0
11). 3y = 10 + 4x
4x = 2y -8
12). 7x = 2y + 6
10x - 2y = 9
13). 4x + y = 21
3x = 12 + 3y
14). 8 = 4x + 2y
x = 2 - 3y
15). 7y = 13 - 6x
3x = 2y + 1
16). 2x - 6 = y
4x = 22 - 3y
17). 10y + 26 = 3x
6 - 5x = 2y
18). 3y = 27 - 8x
2x = 5y + 1
19). 5x = 6 + 6y
2 + 2y = 10x
20). 4y + 3 = 5x
7x - 2y = 6
21). 10 - 2a = 4b
7b + 3a = 16
22). 3 + 2u = 10v
4v + 3u - 5 = 0
23). 5q = 12 - 3p
4p = 6q - 22
24). 7f = 9 - 4g
3g = 6 - 5f
25). 3y - 15 = 6x
2x = 7 - 2y
26). 2x = 3 + 6y
6y = 2 - 8x
27). 9x + 4 = -12y
3x = 12y + 12
28). 4x = 10 - 2y
x = 8 - 2y
29). 11t = 9 + 3s
7t + 6 = -2s
30). 6m = 1 - 2n
9m - 6 = 6n
31). 3i = 5 + 8h
5i - 9 = 12h
32). 8y = 7 - 4z
6y - 41 = 8z
Substitution.
A.
Solve the following equations by substituting one equation into the other.
Level 7/8 Pack 4. Page 7.
1).
y = 2x + 7
y = 3x + 3
2).
y = 4x + 3
y = 2x + 11
3).
y = 5x - 3
y = 3x + 7
4).
y = 4x + 9
y = 7x - 6
5).
y = 4x + 5
y = x + 20
6).
y = 6x - 1
y = 4x + 15
7).
y=x-2
y = 1/2x + 4
8).
y = 1/3x + 3
y = 2/3x - 4
9).
y = 3/4x - 3
y = 1/4x + 5
10). y = 4 - 1/2x
y = 1/2x + 2
11). y = 5 - 2/3x
y = 4 - 1/3x
12). y = 7 - 3/5x
y = 12 - 4/5x
13). u = t - 5
u = 1/3t + 7
14). b = 3/5a + 19
b=a+1
15). q = p - 4
q = 2/7p + 31
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B.
C.
Solve the following equations by substituting one equation into the other.
1).
y=x+ 2
2x + 3y = 26
2).
y=x+1
2x + 2y = 26
3).
x=y-3
2x + 3y = 24
4).
x = 2y - 1
3x + y = 11
5).
y = 2x - 1
8x + 3y = 11
6).
x = 8 - 5y
3x + 2y = 11
7).
y = 6 - 2x
4x + 3y = 22
8).
x = 6 - 8y
3x + 6y = 9
9).
x = 5 + 7y
2x - 4y = 15
10). 3x + 2y = 9
y = 4 - 2x
11). 6x - 3y = 39
x=y+7
12). 2x + y = -13
x = 2y + 1
13). y = 8 - 2x
3x - 2y = 5
14). 3x - 2y = 7
y = 14 - 2x
15). 3n - 4m = 7
m = 45 - 2n
16). a = 3b - 12
2b - 5a = 8
17). x = 6y - 2
8y - 3x = 1
18). t = 1 - 7s
5s - 3t = 10
Rearrange then solve the following equations by substituting one equation into the other.
1).
x-y=1
x+y=3
2).
2x - y = 4
3x + y = 11
3).
2x - y = 5
x+y=4
4).
3x - y = 1
x+y=3
5).
x - 3y = 6
x + 3y = 0
6).
x + 2y = 3
2x + y = 0
7).
x+y=1
3x - 2y = 8
8).
3x - 4y = 1
y - 2x =1
9).
m + 3n = 7
4m - 2n = 7
10). a + b = 3
5a - 5b = 1
11). 3p + 7q = 11
p - q = -4
12). 2d + e = -8
6d - 2e = -27
Trial and Improvement.
Solve the following simultaneous equations by trying different values.
1).
u+v=7
u-v=1
2).
a+b=7
a-b=3
3).
m+n=5
m-n=3
4).
p+q=9
p-q=5
5).
a+b=4
a-b=4
6).
u + v = 12
u-v=6
7).
c+d=9
c-d=7
8).
h + i = 11
h-i=7
9).
r + s = 31/2
r - s = 1 /2
10). p + q = 61/3
p - q = 21/3
11). u + v = 51/4
u - v = 3/4
12). m + n = 7
m-n=2
13). p + q = 2
p-q=4
14). c + d = 3
c-d=7
15). h + i = -3
h - i = -7
16). g + h = -2
g-h=0
17). u + v = -8
u-v=2
18). m+ n = -6
m-n=6
19). r + s = 1/2
r - s = 21/2
20). a + b = -1
a - b = 32/3
Level 7/8 Pack 4. Page 8.
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Simultaneous Linear Equations (Worded Questions).
Forming Equations.
In the following statements there are two unknowns, x and y.
For each question write two equations in x and y. Do not solve them.
1).
2).
3).
4).
5).
6).
7).
8).
9).
10).
11).
12).
13).
14).
15).
The sum of two numbers is 21 and the difference is 7.
The sum of two numbers is 73 and the difference is 11.
Four knives and five forks cost £4.90 in total. Seven knives and three forks cost
£5.70 in total. Let the cost of a knife be x and the cost of a fork be y.
Five nuts and six bolts have a mass of 162 g. Three nuts and two bolts have
a mass of 70 g. Let the mass of a nut be x and the mass of a bolt be y.
Two bowler hats and three berets cost £55. Five bowler hats and two berets cost £88.
Let the cost of a bowler hat be x and the cost of a beret be y.
Edward only has five pence and two pence coins in his pocket.
He has ten coins altogether and their total value is 41p.
Let the number of five pence coins be x and the number of two pence coins be y.
Susan sold 50 tickets for a concert. She sold x £2 tickets and y £5 tickets.
She collected £160.
Beth and Alex have £1.55 between them. Beth has 15p more than Alex.
Let Beth have x p and Alex have y p.
Alec and Ulrika have 22 marbles between them. Ulrika has 8 more marbles
than Alec. Let Alec have x marbles and Ulrika y marbles.
In a netball match between Year 9 and Year 10 there were 34 goals. The Year 10 team
won by 6 goals. Let Year 9 goals be x and Year 10 goals be y.
Bert and Ernie have a combined age of 22. In four years time Bert will be twice
as old as Ernie. Let Bert be x years and Ernie y years.
Sarah and Louise have a combined age of 24. Six years ago Sarah was three times
as old as Louise. Let Sarah be x years and Louise y years.
Jenny and Bill have a combined age of 50. Sixteen years ago Jenny was twice
as old as Bill. Let Jenny be x years and Bill y years.
Keith thinks of two numbers. When he adds the two numbers he gets 8. When he
doubles the first number and takes away treble the second number he gets 1.
Lynne thinks of two numbers. When she doubles the first and adds the second
number to it she gets 15. When she trebles the first number and takes away
double the second number she gets -2 (minus two).
Now go back and solve your equations.
Forming and Solving Equations.
1).
2).
3).
Level 7/8 Pack 4. Page 9.
Two numbers, x and y, have a sum of 53 and a difference of 11.
a). Write two equations in x and y.
b). Solve them to find the values of x and y.
Two numbers, p and q, have a sum of 45 and a difference of 19.
a). Write two equations in p and q.
b). Solve them to find the values of p and q.
John and Andrew have £3.40 between them. John has £1.20 more than Andrew.
John has £u and Andrew £v.
a). Write two equations in u and v.
b). Solve them to find the values of u and v.
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4).
5).
6).
7).
8).
9).
10).
11).
12).
13).
14).
15).
a).
Tessa had £1.00 for snacks. In the school canteen 2 cherry buns and 3 doughnuts
cost £1.20. However 4 cherry buns and 1 doughnut cost exactly £1.00.
a). Write two equations.
b). Solve them to find the cost of a cherry bun and the cost of a doughnut.
Four pigs and three sheep cost a farmer £620. Another farmer pay £630 for
three pigs and five sheep.
a). Write two equations.
b). Solve them to find the cost of a pig and the cost of a sheep.
In a toy box there are blue and green bricks only. Find the weight of each type of
brick if 9 blue bricks and 6 green bricks weigh 324 g and 5 blue bricks and
4 green bricks weigh 200 g.
John says “I’m thinking of 2 different numbers. When I double the first and add it to
the second the total is thirteen. When I double the second and take it away from the
first I get minus one.” What are the two numbers ?
Bill sold 75 tickets for a gig. He sold x £5 tickets and y £8 tickets. He collected £444.
How many of each type of ticket did he sell ?
Three cups and four mugs cost £10.20, whilst two cups and five mugs cost £9.60.
Find the price of a cup and the price of a mug.
Ben thinks of two numbers. When he doubles the first and adds treble the second
number to it he gets 5. When he trebles the first number and takes away double the
second number he gets 14. What are the two numbers ?
The sum of Ann and Fred’s ages is 28. In seven years time Ann will be twice as old
as Fred. How old are they now ?
In a basketball match between the Saints and the Tigers, the Saints won by 21 points.
Altogether 125 points were scored. What was the final score ?
On holiday Lisa finds that the cost of two boat trips and five ice creams would be
£12.90 and the cost of three boat trips and four ice creams would be £16.20.
How much would two boat trips and nine ice creams cost Lisa ?
On a bus trip the fare for 5 adults and 8 children was £17.80.
The fare for 6 adults and 5 children was £16.30 for the same journey.
If a similar trip is made by 4 adults and 7 children what would the cost be ?
Find the values of x and y in each of these rectangles.
(Diagrams not to scale).
b).
c).
d).
5x-2y
2x+y
8 cm
3x-2y
x+2
2 cm
2x-y
2y-6
x
9 cm
10 cm
x+6y
y
4x+3
16). Find the values of x and y in each of these isosceles triangles.
Hence find the size of each angle. (Diagrams not to scale).
a).
b).
c).
2y
x+30
Level 7/8 Pack 4. Page 10.
x+5
y
3x
d).
3y-6
y-10
2x+y
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3x+y
4x-1
x+10
2y+2
x+y-10
2y-5
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Puzzling Algebra (Simultaneous Equations).
The letters or pictures in each row or column add up to the numbers shown.
Write pairs of simultaneous equations where you can. Solve them to find the values of the characters.
Now find the value of the remaining characters and the total represented by the question mark.
1).
3).
d
a
d
b
m
o
m
n
a
a
c
a
c d
b a
c c
a b
15 ?
n p o
n p o
p m m
n p o
27 23
5).
15
13
2).
10
?
19
21
4).
10
6).
u
u
u
w
20
r
s
u
s
u
x
w
v
?
t
r
t
t
15
w
x
u
x
t
r
u
s
?
w
w
u
w
12
s
s
s
r
24
20
21
20 32 36
? 19 23 14
7).
8).
15
23
13
14
?
Level 7/8 Pack 4. Page 11.
11
?
29
36
32
27
18
11 9 19 10
16
?
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9).
a
c
d
b
a
b
d
c
?
a
c
a
b
d
c
a
b
11).
22
18
20
14
16
22
22
22
19
34
?
18 30 22 23
21
18
14
16
14).
24
?
24 16 23 19
?
15).
16).
a c a c a 24
b d d b b 23
e i h g f ?
c a c c c 18
b d b d d 22
20 19 26 24 25
Level 7/8 Pack 4. Page 12.
v w x y
x w x y
v y x w ?
v y v y
32 38 24 33
12).
?
13).
21
10).
29
?
32
24 20 26 34 27
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Inequalities.
A.
Show on a number line all the possible values of:
1).
5).
9).
13).
17).
21).
25).
B.
2).
6).
10).
14).
18).
22).
26).
x < 2
h > 5
m ≤ -6
3 < x < 7
-2 ≥ f > -5
-9 < m ≤ -4
-3 < z ≤ 5
3).
7).
11).
15).
19).
23).
27).
x ≥ -1
y ≥ -2
3 ≤ p
-1 ≤ r <
6 > y >
8 ≥ x >
0 ≥ g ≥
2
-1
-2
-4
4).
8).
12).
16).
20).
24).
28).
x ≤ 3
4 > x
-5 < y
-4 < j ≤ -1
3 > b ≥ -3
-2 > v > -6
-3 ≤ r ≤ -1
The solutions to the following inequalities are whole numbers only (integers).
Write all the possible solutions.
1).
5).
9).
13).
17).
21).
C.
x > 4
d ≤ -3
-4 ≥ r
0 ≤ x ≤ 5
4 ≥ x ≥ 1
-3 ≤ f < 5
-1 > c ≥ -9
0 ≤ x ≤ 3
3 ≥ h ≥ 1
-3 ≤ u < 2
-2 > c ≥ -4
4 ≥ t ≥ 0
-1 ≤ z < 5
2).
6).
10).
14).
18).
22).
2 < x < 5
-1 ≥ p > -5
-6 < k ≤ -4
-1 < b ≤ 4
-2 ≥ d > -8
-7 < n ≤ -4
3).
7).
11).
15).
19).
23).
-3 ≤ t < 1
4 > b > -1
4 ≥ y > -2
0 ≥ a ≥ -2
2 > w > -1
2 ≥ s > -2
4).
8).
12).
16).
20).
24).
-1 < y ≤ 5
4 > v ≥ -3
-1 > x > -6
-3 ≤ h ≤ -1
1 > j ≥ -3
-2 > v > -5
3).
7).
11).
15).
19).
x - 2 ≥ -3
0≥ p-4
5t < 30
-40 ≥ 5g
5 ≤ k
2
5x - 4 ≥ 6
2p + 9 ≤ 5
-5 < 6h + 7
0 < 9 + 3e
b + 11 > 13
4
v + 12 < 6
2
4(3t + 2) ≥ 38
3(7 + 3t) < 12
2e + 1 ≥ 5
3
7j + 3 ≥ 4
6
7d - 2 > 4d + 10
6v + 7 > v + 32
4).
8).
12).
16).
20).
x+4<0
-5 > y - 7
24 > 4g
2j ≤ -6
-4 > c
3
7x + 15 ≤ 50
4t - 5 ≤ 11
-3 > 2f + 7
-4 ≤ 8 + 3d
x-2 ≥1
4
r - 3 < -7
4
3(4y + 7) > 69
-12 < 3(2t - 4)
2 > 3r + 2
10
8 < 5q - 2
11
8u -1 ≤ 3u + 4
6b+ 9> 4b + 1
For each of the following solve the inequality.
Show the solution on a number line.
1).
5).
9).
13).
17).
21).
25).
29).
33).
37).
41).
45).
49).
53).
57).
61).
65).
x+3>7
6>x-1
3x > 18
14 < 7u
v ≥ 4
2
6x - 5 > 31
4t - 5 ≥ 35
5y - 6 ≥ 14
4x - 3 ≤ -7
u + 9 ≥ 12
3
h-3≤1
2
3(2x - 3) ≤ 6
5(f + 4) < 25
a+4>5
2
4 ≤ 5t - 2
7
9h + 3 > 7h + 5
12x + 4 > 7x - 6
Level 7/8 Pack 4. Page 13.
2).
6).
10).
14).
18).
22).
26).
30).
34).
38).
42).
46).
50).
54).
58).
62).
66).
x-5≤1
-3 ≤ t + 2
2h ≤ 22
3f > -12
h < -2
3
3x - 7 < 14
3u - 10 < 14
17 > 3e + 2
5q + 2 > -13
n + 10 < 12
5
c - 2 > -3
3
2(5f - 4) > 27
4(3 + 2g) > 20
h-6<2
5
4p - 7 < 9
5
4c + 9 ≥ 2c + 15
12n + 7 > 5n - 7
23).
27).
31).
35).
39).
43).
47).
51).
55).
59).
63).
67).
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24).
28).
32).
36).
40).
44).
48).
52).
56).
60).
64).
68).
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D.
Solve the following inequalities for x.
(Take care when multiplying or dividing by a negative number).
1).
5).
9).
13).
6 - 2x > 10
4 - 7x ≤ -3
-7x - 3 > 11
6 - x ≤ 12
2
17). 6 ≥ 8 - x
3
21). 3(2 - 3x) < 24
25). 18 > 3(4 - x)
E.
21).
25).
29).
33).
4).
8).
12).
16).
7 - 2x > 13
9 ≤ 19 - 5x
13 - 4x ≤ 5
9 - x < 12
5
20). -4 < -9 - x
5
24). 5(1 - x) < -5
28). 4(3 - 2x) < -4
x2 > 4
v2 < 2.25
4x2 < 36
3y2 > 432
x2 > 12
3
n2 + 3 > 12
14 > u2 - 11
2x2 - 5 > 45
4r2 + 8 > 108
2).
6).
10).
14).
18).
22).
26).
30).
34).
c2 < 9
1 ≥ n2
2x2 > 200
7u2 ≥ 567
c2 < 5
5
c2 - 6 < 30
130 ≤ 9 + v2
5p2 + 3 > 183
1 > 2x2 - 1
3).
7).
11).
15).
19).
23).
27).
31).
35).
b2 ≥ 144
k2 ≤ 0.25
3c2 ≥ 75
144 ≤ 9e2
n2 > 36
4
h2 - 6 > 75
m2 - 19 ≥ 150
3e2 + 2 > 14
427 > 5c2 - 73
4).
8).
12).
16).
20).
24).
28).
32).
36).
m2 ≤ 81
1.44 > j2
64 ≥ 4t2
36 > 9p2
27 > b2
3
2
p + 7 ≤ 11
x2 + 14 < 114
4f2 - 3 > 61
2x2 + 6 > 104
Show on a number line the solution that satisfies each of these inequalities.
1).
5).
9).
13).
17).
21).
25).
29).
33).
G.
4 - 5x ≥ 24
4 > 6 - 2x
9 - 2x ≥ 3
5-x>9
3
19). 13 < 5 - x
4
23). 7(2 - x) ≥ 0
27). -5 ≤ 5(1 - 2x)
3).
7).
11).
15).
Solve the following inequalities. It may help you if you show the answer on a number line.
1).
5).
9).
13).
17).
F.
5 - 3x ≤ 20
12 - 4x ≥ 4
-3 - 9x < -21
3-x<8
4
18). 5 ≥ 11 - x
2
22). 4(5 - 2x) < -28
26). 13 < 2(5 - x)
2).
6).
10).
14).
3<x+3≤7
3≥x+4≥0
1 ≤ 2x - 3 ≤ 5
7 ≥ 2x - 1 ≥ 3
3<x+1<5
2
1<2-x<4
5 < 1 - 2x < 9
19 > 3 - 4x > -1
5<4-x≤7
2
2).
6).
10).
14).
18).
22).
26).
30).
34).
-1 < x - 2 < 3
2 > x - 3 ≥ -2
-4 < 3x + 2 < 5
7 > 3x + 4 ≥ -2
-4 < x - 2 < 0
3
2<3-x≤7
-3 < 3 - 2x ≤ -1
20 ≥ 5 - 3x > 2
-2 < 1 - x < -1
3
3).
7).
11).
15).
19).
23).
27).
31).
35).
-4 ≤ 3 + x < 0
-1 > x - 7 > -4
-7 ≤ 2x+ 3 < -1
-3 > 7x - 3 ≥ 11
-6 ≤ x - 4 < -5
5
-1 < 3 - x < 6
-7 ≤ 3 - 5x ≤ 13
7 > -3 - 5x ≥ -13
-6 < -5 - x < -4
4
4).
8).
12).
16).
20).
24).
28).
32).
36).
1<x+4<7
-2 ≥ x + 1 > -1
3 < 2x + 1 < 7
1 ≥ 5 + 4x >-3
2<x+3<4
4
-2 ≤ 4 - x < 0
-1< 3 - 4x <11
19 > 7 - 6x >1
-3 ≤ -6 - x < 0
3
Here is a mixture of questions. Solve the following inequalities.
1).
5).
9).
13).
17).
21).
25).
a + 4 > -2
2). 3 < 7 + b
7-e ≤3
6). 5 ≥ 2 - f
5i - 4 < 2i + 8
10). 3j + 1 > 5j - 7
2(3n - 4) > 4
14). 4(p - 3) > 8
3(4 - 2s) ≥ 6
18). 2(5 - 4t) ≥ -6
2
w ≤ 225
22). 3x2 ≥ 48
2
a ≤ 27
26). 2b2 < 250
3
5
29). (e - 3)(e + 4) ≥ e + 13
Level 7/8 Pack 4. Page 14.
3).
7).
11).
15).
19).
23).
27).
6c - 4 > 20
4). 9d + 7 > -11
5 - 2g > 11
8). 22 ≥ 7 - 5h
k + 5 ≥ 4k - 7 12). 6m - 4 >8m + 5
-2(3 - 5q) < 4
16). -3(2 - 4r) > 18
-2(3u - 4) < -16 20). 5(3 - 2v) ≥ -15
20 ≤ 5y2
24). z2 - 4 ≥ 140
2
71 ≥ c - 4
28). 2d2 - 3 ≤ 213
3
3
30). (2f - 5)(f - 4) < 52 - 13f
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Inequalities (Worded Questions).
1).
2).
3).
4).
5).
6).
7).
8).
9).
10).
11).
12).
13).
14).
A football stadium can hold 20 000 people, it is always more than half full.
x is the number of people in the crowd.
Write this information in the form of two inequalities.
Club 19-40 Holidays have age restrictions on who can travel (given by their name!).
f is the age of the holiday makers.
Write this information in the form of two inequalities.
Kim shops around for a video game. The most expensive shop sells it for £26.00 whilst in
the cheapest shop it is half this price.
y is the price of the video game.
Write this information in the form of two inequalities.
On a roller coaster ride their are height restrictions. Riders have to be at least 156 cm tall
but less than 2 metres. h is the height of the riders.
Write this information in the form of two inequalities.
William has a maths test out of 10 every week. He always gets half marks or better, but has
never got full marks yet. m is the number of whole marks William scores.
Write this information the form of two inequalities.
A bus can hold 54 passengers sitting down and 22 passengers standing.
d is the number of passengers on the bus.
In the rush hour the seats are always full on the bus.
Write this information for rush hour in the form of two inequalities.
At Kylie’s school there are 8 periods a day. At most she has 3 double lessons in one day.
She never has the same subject twice in a day (unless it is part of the double lesson).
p is the number of different subjects per day.
Write the information about the number of subjects per day in the form of two inequalities.
Ben is 7 years old. Jenny is older than Ben. If Jenny is p years old write an inequality
showing Jenny’s age.
Martha is 24 years old. Susan is at least 4 years younger than Martha. If Susan is k years
old write an inequality showing Susan’s age.
Joe is 4 years older than Mary. Mary is x years old.
a). Write an expression for Joe’s age.
b). Joe is less than 13 years old. Show this as an inequality.
c). Solve this to show an inequality for Mary’s age.
Pam is 7 years younger than Amanda. Amanda is u years old.
a). Write an expression for Pam’s age.
b). Pam is 16 years old or older. Show this as an inequality.
c). Solve this to show an inequality for Amanda’s age.
Luke is three times as old as Chloe. Chloe is h years old.
a). Write an expression for Luke’s age.
b). Luke is older than 12. Show this as an inequality.
c). Show an inequality for Chloe’s age.
Tim is half Jill’s age. Jill is b years old.
a). Write an expression for Tim’s age.
b). Tim is 14 years old or younger. Show this as an inequality.
c). Show an inequality for Jill’s age.
A rectangular garden is 8 metres long and h metres wide.
a). Write an expression for the area of the garden.
b). The area of the garden is greater then 36 m2. Show this as an inequality.
c). Show an inequality for the width of the garden.
Level 7/8 Pack 4. Page 15.
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15). Keith gets paid £2.50 for every fork handle he makes. He makes t fork handles a day.
a). Write an expression for the amount of money he makes a day.
b). He never makes more than £37.50 a day. Show this as an inequality.
c). Show an inequality for the number of fork handles he makes.
16). A builder is to build a house that is 18 metres long and q metres wide.
a). Write an expression for the floor area of the house.
b). Planning permission is granted as long as the floor area of the house doesn’t
exceed (be bigger than) 261 m2. Show this as an inequality.
c). Show an inequality for the width of the house.
17). Kate washes the car for Dad. Every time she washes it she gets paid £4.50.
She washes it g times.
a). Write an expression for the amount of money she makes.
b). She needs to make at least £87.75. Show this as an inequality.
c). Show an inequality for the number of times she needs to wash the car.
d). What is the least number of times she needs to wash the car to make the money ?
18). A plumber charges £12 to come out to visit plus £10 for every hours work.
She works z hours.
a). Write an expression for the amount of money she charges.
b). She guarantees she won’t charge more than £47. Show this as an inequality.
c). Show an inequality for the maximum number hours she thinks she will work.
19). An electrician charges £25 to come out to visit plus £8.50 for every hours work.
He works w hours.
a). Write an expression for the amount of money he charges.
b). He says the job will cost at least £76. Show this as an inequality.
c). Show an inequality for the minimum number hours you would expect him to work.
20). An Internet Service Provider (ISP) charges customers £8.25 a month plus £1.20 for
every hour on the internet. The number of hours is n.
a). Write an expression for the amount of money the ISP charges.
A different ISP just charges for every hour on the internet at £1.50 per hour.
b). Write an expression for the amount of money this ISP charges.
c). Write an inequality showing when the first ISP becomes more expensive than the
second ISP.
d). Solve this. Write a statement about which ISP is the cheaper and when.
21). A mobile phone company charges customers £15 a month plus £4.20 for every hour on the
phone. The number of hours is n.
a). Write an expression for the amount of money the phone company charges.
A different mobile phone company just charges for hourly usage at £5.40 per hour.
b). Write an expression for the amount of money this phone company charges.
c). Write an inequality showing when the first phone company becomes cheaper than the
second phone company.
d). Solve this. Write a statement about which phone company is the cheaper and when.
22). A farmer buys a square field n metres long. He divides it exactly in two and keeps pigs in
one area and sheep in the other.
a). Write an expression for the area of the field the pigs are kept in.
b). This pig area has to be greater than 44 m2. Show this as an inequality.
c). Solve this to show an inequality for the length of the original square field.
23). Freda says “For any value of x, x2 ≥ x.”
Michelle proves it is not true and writes an inequality for when it is wrong.
What is the inequality she writes ?
Level 7/8 Pack 4. Page 16.
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Graphical Inequalities 1 (without equations).
1).
a).
-6
-4
Copy the diagrams onto squared paper, then identify the shaded region in each graph.
b).
c).
6
6
6
4
4
4
2
2
2
-2
2
4
6
-6
-4
-4
-4
4
6
-6
-4
-2
-2
-4
-4
-4
-6
-6
-6
e).
6
6
4
4
4
2
2
2
-2
2
4
6
-6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
h).
6
6
4
4
4
2
2
2
2
4
6
-6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
k).
6
6
4
4
4
2
2
2
2
4
6
-6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
Level 7/8 Pack 4. Page 17.
6
2
4
6
2
4
6
2
4
6
l).
6
-2
4
i).
6
-2
2
f).
6
j).
-6
2
-2
g).
-6
-2
-2
d).
-6
-4
Licensed to Dr Challoner's Grammar School
[email protected]
2).
Here are some more shaded regions to identify. Again, copy the diagrams onto
squared paper. Some may need more than one inequality to identify them.
b).
c).
a).
-6
-4
6
6
6
4
4
4
2
2
2
-2
2
-6
6
4
4
6
-2
-4
-4
-4
-6
-6
-6
6
6
4
4
4
2
2
2
2
-6
6
4
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
6
6
4
4
4
2
2
2
2
-6
6
4
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
j).
k).
6
6
4
4
4
2
2
2
2
4
6
-6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
Level 7/8 Pack 4. Page 18.
6
2
4
6
2
4
6
2
4
6
l).
6
-2
4
i).
6
-2
2
f).
6
h).
-4
-4
-2
-2
-4
-6
-2
g).
-6
2
e).
-4
-6
-2
-2
d).
-6
-4
Licensed to Dr Challoner's Grammar School
[email protected]
Graphical Inequalities 2 (without equations).
1).
a).
-6
-4
Copy the diagrams onto squared paper, then identify the unshaded region in each graph.
b).
c).
6
6
6
4
4
4
2
2
2
-2
2
4
6
-6
-4
-4
-4
4
6
-6
-4
-2
-2
-4
-4
-4
-6
-6
-6
e).
6
6
4
4
4
2
2
2
-2
2
4
6
-6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
h).
6
6
4
4
4
2
2
2
2
4
6
-6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
k).
6
6
4
4
4
2
2
2
2
4
6
-6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
Level 7/8 Pack 4. Page 19.
6
2
4
6
2
4
6
2
4
6
l).
6
-2
4
i).
6
-2
2
f).
6
j).
-6
2
-2
g).
-6
-2
-2
d).
-6
-4
Licensed to Dr Challoner's Grammar School
[email protected]
2).
Draw separate small graphs on squared paper. Label the x and y axes from -6 to 6.
Shade in the region that satisfies these inequalities.
a).
e).
i).
3).
c).
g).
k).
y > -3
y>x+2
y ≥ -1 x - 3
4
d).
h).
l).
x ≤ -1
y ≤ 2x - 1
y<1x+5
6
y≤0
y≥ x
y ≥ -1 x
2
b).
f).
j).
x > -5
y < -3x
y<1x-2
3
c).
g).
k).
y > -1
y>x-1
y≤1x+4
6
d).
h).
l).
x≤ 0
y ≤ -2x + 1
y < -1 x -1
4
y ≤ 2 , y > -3
x≥2 , y> 1
y>x , x≤3
b).
f).
j).
x > -2 , x < 1 c).
y < 3 , x ≤ -1 g).
y < 2x , y > - 3 k).
0 > y > -3
d).
y > -3 , x < 2 h).
y > x , y ≤ - x l).
-5 ≤ x ≤ -1
y≤2 , x<0
y≥1x ,y≥0
2
Draw separate small graphs on squared paper. Label the x and y axes from -6 to 6.
Leave unshaded the region that satisfies these inequalities.
a).
e).
i).
6).
x>2
y < 2x
y<1x+4
2
Draw separate small graphs on squared paper. Label the x and y axes from -6 to 6.
Shade in the region that satisfies these inequalities.
a).
e).
i).
5).
b).
f).
j).
Draw separate small graphs on squared paper. Label the x and y axes from -6 to 6.
Leave unshaded the region that satisfies these inequalities.
a).
e).
i).
4).
y≤4
y ≥ -x
y≥1x
3
y ≤ -1 , x > -3 b).
y ≥ 2x , x > 1 f).
y > 2x - 2 , x ≤ 3 j).
x > -3 , y < 0 c).
y < x + 1 , x ≤ 1 g).
y < 3x , y >-1 k).
1 > y > -4
d).
y > 2 - x , y ≥ 2 h).
y > 2x , y ≤ - x l).
-2 ≤ x ≤ 3
y ≤ -x , x > 0
y ≥ 1x , y ≥ -1
3
Draw separate small graphs on squared paper. Label the x and y axes from -6 to 6.
Leave unshaded the region that satisfies these inequalities.
a).
c).
e).
g).
1≤x≤3 , 2≤y≤5
-3 < x < 0 , -4 ≤ y ≤ 0
y >1 , y < 5-x , y<x+1
y > 0 , y < 2x + 2 , y ≤ 5 - x
b).
d).
f).
h).
3 ≤ x ≤ 5 , -4 ≤ y ≤ 2
1<x<4 , y<x+1 , y>x-3
x≥-2 , y < 4 , y ≥ x-1
y < 2x - 1 , y > 1 , y < 4 - 1 x
2
7).
Draw separate small graphs on squared paper. Label the x and y axes from 0 - 12.
Leave unshaded the region that satisfies these inequalities.
a).
c).
e).
g).
x + 2y ≤ 12 , 4x + y < 12 , x > 0 , y ≥ 0
6x + y < 12, x + y < 8, x > 0, y ≥ 0
2x + y < 10, y < 2x , y ≥ 0 , 4x + 9y < 36
y ≥ 1 x , y < 2x , 3x + 5y < 30
2
Level 7/8 Pack 4. Page 20.
b).
d).
f).
h).
x ≥ 0, y > 0, x + 4y ≤ 8, 5x + 3y < 30
2x + 3y ≤ 18, 3x + y ≤ 12, x > 0, y > 0
y < 3x , y > 0 , 4x + 5y ≤ 40
y ≥ 1 x , 9x + 4y ≥ 36 , 3x + 4y ≤ 36
3
Licensed to Dr Challoner's Grammar School
[email protected]
Graphical Inequalities 1 (with Equations).
1).
a).
Some boundaries have been labelled to help you.
Copy the diagrams onto squared paper, then identify the shaded region in each graph.
b).
c).
6
x=3
6
x = -3 6
4
4
4
2
2
2
y=1
-6
-4
-2
2
4
-6
6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
d).
e).
2
4
6
2
4
6
2
4
6
f).
6
6
6
4
4
4
2
2
2
y=1
-6
-4
-2
2
4
-6
6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
y = -3
-4
-4
-4
-6
-6
-6
g).
h).
-4
6
4
4
4
2
2
2
-2
j).
-6
-4
i).
y=x+5
6
6
-6
x=0
y=x
2
4
-6
6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
k).
6
l).
6
6
y=x-1
4
4
4
2
2
2
-2
2
4
6
-6
-2
-4
-6
Level 7/8 Pack 4. Page 21.
y = -x
y = -x + 2
-4
-2
2
4
6
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
Licensed to Dr Challoner's Grammar School
4
6
y = -x - 4
[email protected]
2).
Here are some more shaded regions to identify. Again, copy the diagrams onto
squared paper. Some may need more than one inequality to identify them.
b).
c).
a).
y = 3x
6
4
-4
-2
2
-6
6
4
-4
-2
2
4
6
-4
-2
2
-2
-4
-4
-4
-6
-6
-6
6
6
4
4
4
2
2
2
2
6
4
-6
-4
-2
2
4
6
-6
-4
-2
-2
-2
-4
-4
-4
-6
-6
-6
h).
i).
6
6
4
4
2
2
-2
2
-6
6
4
-4
-2
-2
6
y=4
y = -1 x + 5
2
4
6
4
6
x=2
4
2
2
4
6
-6
-4
-2
-2
2
-2
y = -3
-4
x = -4
-6
x=1
j).
6
y = -2x
2
-2
g).
4
f).
y = -3x - 3
y = 2x + 2
-2
-4
-6
-2
e).
-4
2
-2
6
-6
y=1x
2
2
d).
-6
4
4
2
-6
6
6
y = -3
-4
-4
-6
-6
k).
l).
6
4
6
6
4
4
2
2
y=3
2
-6
-4
-2
2
4
6
-6
-4
-2
2
4
6
-6
-4
-2
2
4
6
y=0
-2
-2
-2
-4
-4
-4
-6
-6
y = -3
-6
x = -2
Level 7/8 Pack 4. Page 22.
y=x
Licensed to Dr Challoner's Grammar School
y = -x
[email protected]
Graphical Inequalities 2 (with equations).
1).
a).
-6
-4
Some boundaries have been labelled to help you.
Copy the diagrams onto squared paper, then identify the unshaded region in each graph.
b).
c).
6
6
6
4
4
4
2
2
2
-2
2
4
-6
6
-4
-4
6
-4
-2
-4
-4
-6
-6
-6
e).
6
6
4
4
4
2
2
2
2
4
-6
6
-4
-2
2
-2
-2
-4
-4
-6
-6
4
6
-6
-4
-2
y = -x + 4
y = -x - 3
y=x+4
4
2
2
2
-4
-2
2
-2
-2
-4
-4
y = -2x
4
6
4
2
4
-2
6
-6
-4
2
y = -x + 2
-6
6
6
4
4
2
2
-2
2
4
6
-6
-4
-2
-2
-4
-4
-4
Level 7/8 Pack 4. Page 23.
-6
y=x-6
y = -3 x - 3
2
Licensed to Dr Challoner's Grammar School
y = 3x + 9
y = -6x + 30
y = -1x+5
6
2
-2
y = -3x - 3
6
-4
-2
-6
4
l).
y = -1 x + 1
2
2
-4
-6
y = 2x - 3
-2
-6
y=1x-2
2
-2
k).
6
6
6
4
-6
4
i).
4
6
2
-6
6
4
6
-2
6
2
4
-4
h).
-2
2
f).
6
-2
y = -1 x + 3
3
-6
-4
j).
-4
4
-2
-6
-6
2
-2
g).
-6
-2
-2
d).
-6
-4
4
6
y = 1x -3
3
-6
[email protected]
x>4
x≥4
x<4
x≤4
x > -3
x ≥ -3
x < -3
x ≤ -3
x≥0
x≤0
Level 7/8 Pack 4. Page 24.
Licensed to Dr Challoner's Grammar School
[email protected]
4<x
4≤x
4>x
4≥x
-3 < x
-3 ≤ x
-3 > x
-3 ≥ x
0≤x
0≥x
Level 7/8 Pack 4. Page 25.
Licensed to Dr Challoner's Grammar School
[email protected]
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
Level 7/8 Pack 4. Page 26.
Licensed to Dr Challoner's Grammar School
[email protected]
4
2
-6
-4
-2
2
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
4
6
123456789
123456789
123456789
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
4
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
2
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
-4
-2
2
4
123456789012345678901234567890121234
123456789012345678901234567890121234
123456789012345678901234567890121234
-6
6
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
4
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
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2
1234567890123456789012345678901212
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1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
-4 1234567890123456789012345678901212
-2
2
4
6
1234567890123456789012345678901212
1234567890123456789012345678901212
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
12345678901
-6
-4
12345678901
12345678901
12345678901
4
2
-2
2
4
6
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
2 12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
12345678901234567890123
2
4
6
12345678901234567890123
12345678901234567890123
4 12345678901234567890123
12345678901234567890123
-6
-4
-2
Level 7/8 Pack 4. Page 27.
4
2
-6
-4
-2
2
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
123456789
4123456789
6
123456789
123456789
123456789
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
4
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
2
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
-6
-4
-2
2
4
1234567890123456789012345678901212345
1234567890123456789012345678901212345
1234567890123456789012345678901212345
-6
6
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
4
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
2
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
1234567890123456789012345678901212
-4 1234567890123456789012345678901212
-2
2
4
6
1234567890123456789012345678901212
1234567890123456789012345678901212
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
123456789012
-6
-4
123456789012
123456789012
123456789012
4
2
-2
1234567890123456789012
1234567890123456789012
1234567890123456789012
4
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
2
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
1234567890123456789012
-6
-4
-2
1234567890123456789012
1234567890123456789012
Licensed to Dr Challoner's Grammar School
2
4
6
2
4
6
[email protected]
-3 ≤ x ≤ 4
-3 ≤ x < 4
-3 < x ≤ 4
-3 < x < 4
4 ≤ x ≤ -3
4 < x < -3
0 ≤ x < -3
-3 ≤ x ≤ 0
0≤x<4
4<x≤0
Level 7/8 Pack 4. Page 28.
Licensed to Dr Challoner's Grammar School
[email protected]
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
Level 7/8 Pack 4. Page 29.
Licensed to Dr Challoner's Grammar School
[email protected]
-6
-6
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
4
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
2
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
-4 12345678901234567890123456
-2
2
4
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
12345678901234567890123456
4
12345678901234567890123456
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12345678901234567890123456
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12345678901234567890123456
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Licensed to Dr Challoner's Grammar School
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Translations.
Copy all of the following diagrams onto squared paper.
Each grid has an x-axis from -4 to 4, and a y-axis from -7 to 7.
Write the vector that describes the translation from A to A' and B to B' for each question.
1).
2).
3).
4).
y
y
y
y
A.
A'
A
A'
A
B
A
A'
x
x
x
x
A
B
B'
B'
B
A'
B'
B
B'
y
5).
y
6).
A'
y
7).
8).
y
A
A'
A
B
A
B
A
x
x
B'
x
B
x
A'
B'
B'
B.
B
A'
B'
Translate the shape through the vectors shown, starting from the original diagram each time.
Draw these translated shapes onto your squared paper.
Write the coordinate of the point X after each translation.
1).
a).
y
( 51 )
b).
( 09 )
c).
4
6
4).
a).
2).
a).
b).
x
( )
y
1
-4
( )
c).
5).
a).
b).
(-4-2 )
b).
c).
-3
(-11
)
x c).
Level 7/8 Pack 4. Page 31.
y
( 19 )
(-50 )
x
-4
7
( )
y
-1
5
( )
(-64 )
(-41 )
Licensed to Dr Challoner's Grammar School
y
3).
a).
( 60 )
b). ( 4 )
-6
c). ( 1 )
-10
6).
a).
b).
x c).
x
y
-3
2
( )
(-2-5 )
x
( -5-8 )
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Reflections.
A.
Copy all the diagrams onto squared paper. The equation of the mirror line is given in bold.
Draw the mirror line. Reflect the shapes in the mirror line.
Write the coordinate of the point marked x after each reflection.
y
y x=1
y y = -2
y y=1
1).
2).
3).
x=0
4).
x
5).
y
y=x
x
6).
y
y = -x
x
7).
y y = x+1
x
x
x
8).
y
x
y = -x-3
x
The following diagrams are sets of objects and images. Each set has a different mirror line.
Draw the mirror line. Write the equation of the mirror line.
B.
1).
y
2).
y
x
5).
y
Level 7/8 Pack 4. Page 32.
y
x
6).
x
3).
y
4).
y
x
7).
y
x
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x
8).
x
y
x
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Rotations.
Copy all of the following diagrams onto squared paper.
Rotate each diagram about the point marked , through the given degrees and direction.
Write the coordinate of the point marked x after each rotation
Rotate 90˚ Anticlockwise for Questions 3 and 4.
A. Rotate 90˚ Clockwise for Questions 1 and 2.
y
y
y
y
1).
2).
3).
4).
x
x
Rotate 180˚ for Questions 5 and 6.
y
y
5).
6).
1).
x
Rotate 270˚ Clockwise for Questions 7 and 8.
y
y
7).
8).
x
x
B.
x
x
x
Describe fully the rotations from A to A' and B to B' in each diagram.
y
y
2).
3).
4).
y
y
B'
A
A'
A' A
A'
x
A
A'
x
B'
x
B
B
x
B
B'
A
B
B'
5).
y
y
6).
y
7).
B'
A
B
x
B'
A
B
A'
x
x
x
A
A'
Level 7/8 Pack 4. Page 33.
A'
B'
B'
A
A'
y
8).
B
B
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Enlargements.
le
Sca ors
t
Fac
A.
Copy the following diagrams onto squared paper.
You should be able to fit 2 questions on one side of A4.
Each grid has an x-axis from -6 to 6, and a y-axis from -9 to 9.
Write the scale factor and centre of enlargement
that describe the enlargements A to A' and B to B' for each question.
y
1).
y
2).
y
3).
A
A
A
A'
A'
A'
x
B
x
x
B'
B'
B
B'
B
y
4).
y
5).
y
6).
A'
A
A'
A
A
B'
x
A'
x
B
x
B'
B
B'
B
y
7).
y
8).
y
9).
A'
A
A'
A
A'
A
x
x
x
B'
B'
B'
B
B
B
Level 7/8 Pack 4. Page 34.
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B.
Put each shape through the enlargement described.
Write the coordinate of the point x after each enlargement.
1).
2).
3).
A- Scale factor 3, about (5,7)
A- Scale factor 2, about (-4,9) A- Scale factor 3, about (1,6)
B- Scale factor 2, about (-6,-9) B- Scale factor 5, about (0,-2) B- Scale factor 2, about (6,3)
y
y
y
A
A
A
x
x
x
B
B
B
4).
A- Scale factor 1/3, about (6,0)
B- Scale factor 1/2, about (-6,1)
y
5).
6).
1
A- Scale factor /5, about (0,4) A- Scale factor 1/5, about (1,9)
B- Scale factor 1/2, about (-6,-9) B- Scale factor 1/3, about (1,-3)
y
y
A
A
A
x
x
x
B
B
B
7).
A- Scale factor 3, about (5,1)
B- Scale factor 1/2, about (-5,2)
8).
A- Scale factor 1/3, about (0,6)
B- Scale factor 2, about (-6,0)
y
9).
A- Scale factor 3, about (4,2)
B- Scale factor 1/4, about (0,-5)
y
y
A
A
A
x
x
x
B
B
B
Level 7/8 Pack 4. Page 35.
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Constructing Enlargements.
Construct the enlargement for these objects.
Use the given scale factor and centre of enlargement marked x.
D.
S.f. 3
B.
S.f. 3
D.
A.
B.
C.
A.
S.f. 2
C.
S.f. 2
E.
S.f. 1/2
G.
F.
S.f. 3
E.
F.
S.f. 1/3
G.
I.
I.
S.f. 4
H.
S.f. 3
M.
K.
J.
1
S.f. /2
S.f. 2
K.
S.f. 2
M.
N.
N.
S.f. 1/2
L.
L.
S.f. 4
Level 7/8 Pack 4. Page 36.
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Transformations.
Copy the diagrams onto squared paper.
Describe fully the transformation from the object (ABCD) to the image (A'B'C'D').
2).
3).
1).
y
y
y
C
Im.
Im.
Obj.
B
x
D
x
A
x
B'
C'
Obj.
A'
D'
4).
5).
6).
y
y
D'
C'
A'
B'
D
A
y
Obj.
x
C
B
Obj.
x
x
Im.
7).
Im.
8).
9).
y
y
y
C
Im.
x
Im.
x
D
B
x
A
A'
B'
Obj.
Ob.
10).
D'
11).
y
Im.
Ob.
x
12).
y
Level 7/8 Pack 4. Page 37.
y
Obj.
x
Im.
D'
A'
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C'
D
C
A
B
x
C'
B'
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These get harder, be careful.
13).
14).
15).
y
y
y
D
B
A
C
D
B'
C'
A'
D'
A
x
16).
C
D'
A'
C'
B'
B
x
17).
B
B'
A'
C'
D'
B'
A'
C'
D'
B
B
D
C
B
C
B'
x
A'
A
D
x
D'
C'
20).
21).
y
y
C
D
A
x
C
A
x
19).
A'
D'
y
D
B
A
A'
y
C
A
C'
18).
y
D
B'
D'
C' D
A'
B'
y
C
Obj.
x
x
x
B'
Im.
D'
A
C'
22).
B
23).
24).
y
y
y
B
C
B
D
A
A
x
C
A'
x
Obj.
x
Im.
D'
C'
C'
A'
B'
B'
Level 7/8 Pack 4. Page 38.
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Combining Transformations.
A.
The transformations used in these problems are
S
U
W
a reflection in the x-axis
T
a reflection in the y-axis
a reflection in the line y = x
V
a reflection in the line x = 3
a reflection in the line y = -4
Note: VT(A) means transform T(A) through transformation V.
Copy each diagram and answer the questions.
1).
On the diagram draw the position of
2).
a). T(A)
b). VT(A).
c). What single transformation maps
A directly on to VT(A) ?
On the diagram draw the position of
a). W(B)
b). SW(B).
c). What single transformation maps
B directly on to SW(B) ?
y
y
A
x
x
B
3).
On the diagram draw the position of
4).
a). V(C)
b). SV(C).
c). What single transformation maps
C directly on to SV(C) ?
On the diagram draw the position of
a). W(D)
b). TW(D).
c). What single transformation maps
D directly on to TW(D) ?
y
y
C
x
x
D
5).
On the diagram draw the position of
6).
a). U(E)
b). SU(E).
c). What single transformation maps
E directly on to SU(E) ?
On the diagram draw the position of
a). T(F)
b). UT(F).
c). What single transformation maps
F directly on to UT(F) ?
y
y
E
x
x
F
Level 7/8 Pack 4. Page 39.
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B.
Copy each diagram and answer the questions. The transformations used in these problems:
S
U
W
Y
1).
an enlargement s.f. 4 about (-6,-6) T
an enlargement s.f. 1/2 about (-6,-6) V
a rotation 180˚ about (-2,-2)
X
a rotation 180˚ about (-1,4)
Z
On the diagram draw the position of
2).
a). S(A)
b). US(A)
c). What single transformation maps
A directly on to US(A) ?
an enlargement s.f. 1/4 about (-6,-6)
a rotation 180˚ about (0,0)
a clockwise rotation 90˚ about (0,0)
a clockwise rotation 90˚ about (2,2)
On the diagram draw the position of
a). T(B)
b). ST(B).
c). What single transformation maps
B directly on to ST(B) ?
y
y
x
x
B
A
3).
On the diagram draw the position of
4).
a). V(C)
b). WV(C).
c). What single transformation maps
C directly on to WV(C) ?
On the diagram draw the position of
a). Y(D)
b). VY(D).
c). What single transformation maps
D directly on to VY(D) ?
y
y
D
C
x
x
5).
Write your findings about two half turns about different centres of rotations.
Hint. Join up the 2 centres and join up 2 corresponding points on the rotated shapes.
6).
On the diagram draw the position of
7).
a). X(E)
b). ZX(E).
c). What single transformation maps
E directly on to ZX(E) ?
On the diagram draw the position of
a). Z(F)
b). WZ(F).
c). What single transformation maps
F directly on to WZ(F) ?
y
y
E
x
Level 7/8 Pack 4. Page 40.
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F
x
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Transformation Investigations.
1).
Paper.
A4 paper is part of a series of paper sizes.
A4 paper is 297 x 210 mm. Find all the other paper sizes in this range.
Plot a graph of the height against width for the paper sizes, what do you notice ?
The area of A5 paper is half the size of the area of A4 paper, and the area of A3 paper is
twice the area size of A4 paper.
Find the area scale factor between the consecutive ‘A’ sizes.
Find the length scale factor between the consecutive ‘A’ sizes.
What is the link between the area scale factor and the length scale factor for ‘A’ paper ?
How many A5 sheets can be cut from one A4 sheet of paper ?
How many A6 sheets can be cut from one A5 sheet of paper ?
How many A6 sheets can be cut from one A4 sheet of paper ?
Explore all the possibilities.
A0 is the largest paper size, what is the area of this paper ?
70 gram paper means a square metre weighs 70 grams.
What would be the weight of one sheet of A4 paper made from 70 gram paper ?
2).
Cube.
For a cube of side length 1 cm, find the surface area and volume.
The side length is enlarged by a scale factor 2.
Find the surface area and volume of the enlarged cube.
Vary the scale factor by which the side length is enlarged.
Find the surface area and volume of the cube in each case. Record your results.
For a cuboid of length 4 cm, width 2 cm and height 3 cm find the surface area and volume.
The sides are enlarged by a scale factor 2.
Find the surface area and volume of the enlarged cuboid.
Vary the scale factor by which the side length is enlarged.
Find the surface area and volume of the cuboid in each case. Record your results.
Find a link between the length scale factor, the area scale factor and the volume scale
factor when a solid is enlarged.
A cuboid has a length 50 cm, a width 30 cm and a height 20 cm.
Find the surface area and the volume of this cuboid.
Find the surface area and volume of the cuboid if you
a). double the length,
b). double the length and width,
c). double the length, width and height.
What are the relationships between the original and the enlarged surface area and volume ?
Level 7/8 Pack 4. Page 41.
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3).
Combinations.
I is the Identity transformation.
J is a reflection in the x-axis.
K is a reflection in the y-axis.
L is a rotation 90˚ clockwise about (0,0).
M is a rotation 180˚ about (0,0).
N is a rotation 90˚ anticlockwise about (0,0).
Investigate the effect of pairs of transformations on any given shape.
Does the order in which these transformations are applied to the given shape matter ?
4).
Kaleidoscope.
Draw two mirror lines that are not perpendicular to each other.
Investigate the reflections in the mirror lines.
Change the number of mirror lines.
How many mirror lines are needed so that the reflections can be represented by
a single rotation ?
5).
Linkages.
Some transformations can be reproduced using linkages.
To make these linkages you will need two different lengths of cardboard strip (or
Geostrip), one length twice the length of the other. Along with plenty of cardboard strips
you will also need butterfly clips and drawing pins.
Drawing
pin
Make these linkages.
Which transformation do these linkages reproduce ?
Find other linkages that produce this transformation.
Find linkages that produce a translation.
Find linkages that produce an enlargement.
Investigate linkages.
..and for fun.
Maths
Mystery
Ma
My ths
ster
y
6).
Maths Mystery
Rotate the book through an angle of 180˚ and then rotate it again through 180˚ so the
book ends up at 90˚ to its original position.
Level 7/8 Pack 4. Page 42.
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