Transcript x + 2 - Wrenn Academy

Wrenn Academy Year 11
Mathematics Revision Session
Paper 1
Wednesday 25th May 2016
Number Topics
Estimate the value of:
Answer
79.7
2.13 x 7.85
80
2x8
80
=5
16
Round each
number to
1
significant
figure!
Estimation and Place Value
Use approximations to estimate the
value of –
316 x 4.03
0.198
You must show your working
First make sensible approximations
300 x 4
0.2
Then work out 1200 ÷ 0.2
12 000 ÷ 2 = 6000
Using the information that
14 x 23 = 322
Write down the value of..
(i) 1.4 x 2.3
1.4 x 23 = 32.2
1.4 x 2.3 = 3.22
Answer
(ii) 322 ÷ 2.3
322 ÷ 23 = 14
322 ÷ 2.3 = 140
This
Re-arrange
Notice
that
question
has
the
original
the
decimal
both
values
calculation
point
with has
one
first!
moved
the
decimal
other
way
point
when
you
movement
divide!
Using the information that
58 x 117 = 6786
Write down the value of..
(i) 0.58 x 117,000
5.8 x 117 = 678.6
0.58 x 117 = 67.86
0.58 x 117,000 = 67,860
Answer
(ii) 67.86 ÷ 5.8
6786 ÷ 58 = 117
67.86 ÷ 58 = 1.17
67.86 ÷ 5.8 = 11.7
Notice
which way
the
decimal
point
moves!
Write each of these
numbers correct to 1
significant figure..
a) 26,366
All answers
have only
one digit
other than
0!
30,000
b) 0.0004349 0.0004
Answer
c) 45,071
50,000
d) 0.050869 0.05
Factors, Multiples, HCF and LCM
Find the Highest Common Factor (HCF) of
54 and 135
Prime factors
of 54 = 2x27 = 2x3x9 = 2x3x3x3
of 135 = 3x45 = 3x3x15 = 3x3x3x5
Common factors are 3x3x3
Highest Common Factor = 27
Find the Least Common Multiple (LCM) of
28 and 42
Multiples of 28: 28; 56; 84; 112; 140; …………
Multiples of 42: 42; 84; 126; ………
Least Common Multiple = 84
The cost of a CD player
is £200 + vat. The rate
of vat is 17.5%.
What is the cost of the
CD player?
VAT is easy
Answer
10% of 200 = 20
5% of 200 = 10
2.5% of 200 = 5
35
200 + 35 = £235
to work out
by hand if
you do the
following..
Put these in order of size,
starting with the smallest;
1
4
Answer
0.3
28%
0.25 0.3
0.28
2
= 4
5
10
0.4
Now sort
First
turn
them
these values
starting
with
into decimals
the
smallest
When multiplying
fractions, just
multiply the top
two numbers then
the bottom two
numbers
Work out:
1
2
4
x
= 4
5
10
Give your answer in its
simplest form
Answer
4
10
=
2
5
Now
simplify
it!
Work out:
Answer
1
8
3
+
4
1
8
6
8
+
7
=
8
When adding
fractions you have
to make the
denominators
(bottom numbers)
the same!
Work out:
Answer
To divide fractions,
turn the second
fraction upside down
(invert) and then
multiply the fractions
2
3
÷
2
3
2
4
÷x 4 =
12
1
4
You can leave
your answer
like this or
simplify it if
asked!
Work out:
1
2
3
5
3
Answer
÷1
x÷
25
27
4
5
9
5
5
9
First convert the
mixed numbers to
top heavy
(improper or
vulgar) fractions
Now deal
with it the
same way as
question 3!
Ratio and Proportion
Mrs Jones inherits £12 000
She divides the £12 000 between her three children Laura,
Mark and Nancy in the ratio 7:8:9
How much does Laura receive?
Laura, Mark and Nancy receive 7+8+9 shares
(24 shares) between them
One share is worth £12 000 ÷ 24 = £500
Laura receives 7 shares
Laura receives £500 x 7 = £3500
If 3 coffees cost £4.17,
what would 7 coffees cost?
4.17 ÷ 3 = £1.39
£1.39 x 7 = £9.73
Answer
One
coffee!
Similar means
These two triangles are similar.
that one
is an
Find the lengths of x andtriangle
y
enlargement
of the other
20 cm
Answer
30 cm
y
x
12 cm
5 cm
30 ÷ 12 = 2.5 scale factor
So..y = 5 x 2.5 = 12.5cm
And..x = 20 ÷ 2.5 = 8cm
Standard Form
Here are six numbers written in standard form.
2·6 x 105 1·75 x 106 5·84 x 100 8·2 x 10-3
3·5 x 10-1 4·9 x 10-2
2.6 x 105 = 260 000
1.75 x 106 = 1 750 000
5.84 x 100 = 584
8.2 x 10-3 = 0.0082
3.5 x 10-1 = 0.35
4.9 x 10-2 = 0.049
Write down the largest number
1.75 x 106
Write down the smallest number
8.2 x 10-3
3
10
a) Write 2.7 x
ordinary number
as an
.
2 7 00 0
b) Write 3.12 x 10-3 as an
ordinary number
Answer
.
0 00 3 1 2
a) Write 38,500,000 in
standard form
3.85 x 107
b) Write 0.000005 in
standard form
Answer
5 x 10-6
Standard
form must
start with
values
between 1
and 10!
Algebra Topics
Substitution
Find the value of
This is a SUBSTITUTION question
3x + 4y
If x = 6, 3x = 3 lots of 6
When x = 6 and y = -3
And if y = -3, 4y = 4 lots of -3
So … 3x + 4y
= (3 x 6) + (4 x -3)
= 18 + -12
Adding a negative number means you
take it away
So the answer is 6
Find the value of 5p + 2q when p = 4 and q = -7
5p + 2q will be (5 x 4) + (2 x -7)
This is (20) + (-14)
Adding a negative number is the same as taking it
away
So 20 – 14 = 6
Find the value of u2 – v2 when u = 5 and v = 3
u2 – v2 will be 52 – 32
This is 25 – 9
So the answer is 16
Simplifying Simplify
Expressions
5x + 3y – 2y + 4y
Simplify means collect together like terms,
So that’s all the xs and all the ys
It can be easier to do the adding first
So … 5x + 3y + 4y – 2y
So that gives …
5x + 7y – 2y
= 5x +5y
Simplify the expression…
2
5p +
Answer
3q –
2
p
4p2 + 5q
+ 2q
Expanding and Simplifying
Expand and simplify 4(m + 3) + 3(2m – 5)
First multiply out the brackets
4m + 12 + 6m – 15
Then simplify
10m - 3
Solving Equations
7z + 2 = 9 – 3z
7z + 3z = 9 – 2
Watch out
here with
signs
10z = 7
z = 7 ÷ 10
z = 0∙7
Sort out the
7z and 3z and
the numbers
first
Solve the equation…..
4(3n + 7) = 16
12n + 28 = 16 - 28
Brackets
Answer
12n = -12
It is ok to
have a
negative
answer!
n = -1
first!
Solve the equation…
Answer
4(x + 2) = 6x + 4
4
-4x
4 + 8 = 6x -+ 4x
4 = 2x
x=2
Brackets
first, then
group the
terms!
Solve these equations
6r + 2 = 8
6r = 8 – 2
6r = 6
Be careful here!
r = 1
7s + 2 = 5s + 3
7s – 5s = 3 – 2
2s = 1
s = ½
12- y = 5
3
And then,
of course,
there’s here!
And here!
12 – y = 5 x 3
12 – y = 15
12 – 15 = y
And especially here!
-3 = y (which is the same as y = -3)
Solve 4(z - 1) = 2(z + 3)
Multiply out the brackets first ..
4z - 4 = 2z + 6
Then solve as before – careful with the signs!!!
4z – 2z = 6 + 4
2z = 10
z = 5
Expand and simplify (4x - 3)(x + 5)
This is harder. Multiply everything in the first bracket by
everything in the second bracket
4x2 + 20x – 3x – 15
Simplify
4x2 + 17x - 15
Solve this
simultaneous equation
Multiply the
first equation
by 3 to make
the x values
the same
3x
x ++ 3y
9y == 13
39
Answer
3x + 2y = 4
7y = 35
y=5
x + 3(5) = 13
x = -2
(x3)
Now subtract (there are
no x’s left then!)
Now
substitute
y=5 into
the first
equation
x + 3y = 7
Ooh
look!
Two 6’s
+
(x 2)
Give some values
ofWhat
x and y that
2x + 6y = 14
make
does
this this
3x - 6y = 6
equation
true
word
mean??
What should
How can I change
5x + 0ythe
= first
20
you do now?
equation to make the x or y
Hint: if I added them,
These
equations
Co-efficientswhat
the would
happen?
are simultaneous.
same?
What
So…. x = 4…how do we find
y? does this
mean?
Expanding Brackets
Expand
4x(x2 + 5)
Expand means multiply out the brackets
That means mutliply both terms inside
the bracket by 4x
So the answer is
4x3 + 20x
Expand these brackets……….
Answer
a) 7(x + 3)
7x + 21
b) x(x + 3)
x2 + 3x
c) 2y(3y – 5)
6y2 – 10y
Remember to
multiply out
both terms!
Multiply out
the brackets
first!
Expand and simplify……
2(4a + 2) – 3(2a – 4)
8a + 4 – 6a + 12
Answer
2a + 16
Notice that this
sign has changed to
a +, because of the
– outside of the
brackets
Expressions 2
Simplify……
i)
2
3a b
x
3
7ab
21a3b4
(x ++ 2)
2)2
ii) (x
Answer
(x + 2)
(x + 2)
Notice how
the powers
have
changed!
Expand these brackets……….
Answer
a) (x + 1)(x + 3)
x2 + 3x + 1x + 3
x2 + 4x + 3
b) (x – 6)(x + 2)
x2 + 2x - 6x - 12
You should get 4
2
x - 4x - 12
terms when you
multiply out the
c) (x – 4)(x + 7)
brackets!
x2 + 7x - 4x - 28
x2 + 3x - 28
Factorise
Factorising
2x + 6
This means take out the factors that are
common to both terms.
So that would be 2, giving an answer of
2(x + 3)
Factorise x2 + 6x
Here the x is common to both terms,
giving an answer of
x(x + 6)
Factorise…..
12x + 4
Find the
common
factor(s) in all
terms. In this
case it is 4!
12x
4 +4
(3x
1)
Answer
Notice that this
is 1 and not 0!
Factorise ….
6x + 18y
6
(x + 3y)
6
6x
18y
Answer
Find the
common
factor(s) in all
terms. In this
case it is 6!
Factorise……
8xy + 12x
Find the
common
factor(s) in all
terms. In this
case it is 4x!
4x
8xy
4x
(2y + 12x
3)
Answer
Why is this
just 3?
Factorise…..
6x2 – 3xy
2 – y)
6x
(2x
3x
3xy
3x
Answer
Find the
common
factor(s) in all
terms. In this
case it is 3x!
Factorise……….
Answer
Find the
common factors
in all the terms
a) x2 + 2x
x (x + 2)
b) y2 – 6y
y (y – 6)
c) 8x2 – 20xy 4x (2x – 5y)
Factorise……….
This means
separate into
two brackets!
a) x2 + 3x + 2 (x + 1)(x + 2)
b) x2 + 7x + 12 (x + 3)(x + 4)
c) x2 + 2x - 15 (x + 3)(x - 5)
d) x2 - 2x - 35 (x + 5)(x - 7)
Answer
The number values in each set of
brackets must multiply together
to make the 3rd term in the
quadratic, and must add together
to make the number of x’s in the
2nd term!
Plotting Straight Line Graphs / Curved Graphs
Complete this table of values for y = 2x - 1
-1
3
To get the y value, you double the x value and then take
away 1
So when x = 0, double 0 is still 0, take away 1 = -1
When x = 2, double 2 = 4, take away 1 = 3
A useful check that you are right, is to notice that this
equation gives a straight line, so there will be a pattern in
the y numbers.
There should be the same difference between all of them.
Here the difference is 2
On the grid, draw and label the lines y = -4 and y = 2x + 1
The point y = -4 is here.
x
x
So the line y = -4 is the line
where all the
y coordinates are -4
To find the line y = 2x + 1
Substitute some values for x to
get y
x
If x = 0, 2x + 1 = 1, y = 1
If x = 1, 2x + 1 = 3, y = 3
If x = 2, 2x + 1 = 5, y = 5
y = -4
And so on
Join the dots!
Write down the point where the
lines y = -4 and y = 2x + 1 cross
(-2∙5, -4)
Complete the table of values for y = x2 - 3
x
-2
-1
0
1
2
3
y
1
-2
-3
-2
1
6
When you see x2 you expect
to see a curve. You also
expect to see a symmetrical
number pattern in the table.
Work out the positive values
first
x When y = 2, y2 will be 4, take
away 3 will be 1
When y = 3, y2 will be 9, take
away 3 will be 6
x
x
x
x
x
When y = -1, y2 will be 1,
(NOT -1!!) take away 3 will be
-2
Then, on the grid, draw
the graph of y = x2 - 3
Plot the points and join
with a curve
Representing and Solving Inequalities
Solve this in the same way as you would an equation
3x + 8 < 29
So 3x < 29 – 8
3x < 21
x < 21 ÷ 3, so x < 7
Write down the inequality shown by the
following diagram
This shows that x is less than 2.
The empty circle means 2 is not included
So x < 2
Solve the inequality…
Solve it like
an equation
2x + 7 > 1
2x + 7 > 1 - 7
Answer
2x > -6
x > -3
Write down all the values
of m if -2 < m ≤ 4 and m
is an integer……
7
6
5
4
3
2
1
0
Answer
-1
-2
-3
-4
-5
m can be any of
these numbers,
but not -2
Write down all the possible
values of p when -3 ≤ p < 2
and p is an7 integer….
6
5
4
3
2
1
Answer
0
-1
-2
-3
-4
-5
p can be any of
these numbers,
but not 2
This diagram shows an inequality
-3
-2
-1
0
1
2
3
4
5
6
7
x
We can write the inequality using symbols
How?
-3
-2
-1
0
1
2
A filled in spot
means that x
includes that
number
3
4
x is somewhere
between the
spots
5
6
7
An empty spot
means that x
DOESN’T include
that number
So…..
-3
-2
-1
0
1
2
3
4
5
6
7
In words…..
x is more than or equals -2
and is less than 5
In symbols…..
x ≥ -2 and x < 5
Which is written as
-2 ≤ x ‹ 5
Shape and Space Topics
Transformations
The diagram shows two identical shapes, A and B
Imagine a hook attached
at one end to (0,0) and
the other end to a
corner of shape A
If this was rotated half
a turn anticlockwise,
shape A would move to
position B
So the single
transformation taking
shape A to shape B is a
rotation of 180º
anticlockwise about (0,0)
Describe fully the single transformation which takes
shape A to shape B
AREA OF COMPOUND SHAPES -A shape has
dimensions as shown
First,
split this
shape
into
smaller
shapes
that we
recognise
Area of triangle A
A
B
b x h
2
C
3 x 4
2
= 6 cm2
Calculate the shaded area
Area of rectangle C
l x w
= 10 x 1
= 10 cm2
Total area = 6 + 4 + 10 = 20 cm2
Area of triangle B
b x h
2
2 x 4
2
= 4 cm2
SURFACE AREA AND VOLUME
The area of the cross section of the triangular prism is
20cm2. The length is 12cm
Work out the volume of the prism
Answer
20cm2
Volume = 20 x 12
Volume = 240cm3
Volume = cross
sectional area x
length
What do the external angles total?
What is the value of x?
x
x
x
The external
angles of a
polygon
always add up
to..360o
360 ÷ 5 = 720
Answer
x
x
The exterior angle of a regular polygon
is 45o . How many sides has the polygon?
45o
45o
45o
45o
45o
Answer
45o
45o
45o
The polygon has 360 ÷ 45 = 8 sides
Find the value of x
107
o
34
o
48
63
o
x
Answer
47
o
107 + 63 + 47 + 34 + 48 + x = 360
x = 61o
o
Find the size of the interior angle of
this regular octagon
So the
interior
Find
the
angle
must
exterior
be??
angle
first!
Answer
o oo
360 ÷ 8 = 45
135
45
180 – 45 = 135o
Angle Facts
The diagram shows a regular octagon
Calculate the size of the exterior angle of the regular octagon,
marked y on the diagram.
We know this is a REGULAR octagon, so all the exterior angles
are equal
The total of all the exterior angles is 360º (a full turn)
So angle y will be 360 ÷ 8
Angle y = 45º
Work out the size of angles x and y.
Give reasons for your answers x = 125
125
Corresponding or
F angles
o
125
x
o
y
55
Answer
0
o
y = 550
Angles on a
straight line =
1800
Work out the size of angles x and y.
Give reasons for your answers
62
o
40
o
o
118x
o
62y
y = 62o
Alternate or
Z angles
Answer
x = 118o
Notice the inside
angles are
complementary
Find the angle marked g. Give a reason
for your answers
Work out this angle
first (angles on a
straight line)
28
o
o
57
o
95
o
g
57
Answer
g = 570
Alternate or Z
angles
Find angles x and y.
Give reasons for your answers
68
o
o
o
60
y
52
o
120
68
x
x is 68o
Z angles
o
Answer
600
Angles on a
straight line
y = 520
Angles in a
triangle
Handling Data Topics
Pie Charts
The holiday destinations of 30 students were recorded
Draw a clearly labelled pie chart to represent this information
First, work out the sizes of the angles that will represent each
destination
There are 360° in a circle and there
are 30 students, so each student will
be represented by 12° (360 ÷ 30)
America
France
France will be 6 x 12 = 72°
Greece
Spain will be 9 x 12 = 108°
Spain
Italy will be 3 x 12 = 36°
Greece will be 7 x 12 = 84°
Italy
America will be 5 x 12 = 60°
STEM and LEAF -The stem and leaf diagram shows the ages, in years, of 15 members
of a badminton club
(a) What is the median age of the members
The median is the middle value, when the values are in order
The middle value of 15 will be the eighth value
This is 42
(b) What is the range of their ages
The range is the difference between the youngest and the oldest
member.
So that would be 62 – 27 = 35 years
PROBABILITY - Tom and Sam take turns to throw a dart at a
target
The probability that Tom hits the target is 0·3
The probability that Sam hits the target is 0·2 Complete the
tree diagram
0·8
0·2
0·7
0·8
Probability
The diagram shows a spinner. When the arrow is
spun, the probability of scoring 2 is 0∙3.
The arrow is spun twice and the scores are added.
(a) Complete the tree diagram
The probabilities must add up to one
0∙3
0∙3
0∙7
2
2
0∙7
3
0∙3
2
3
0∙7
To get a 4, you
need to score 2
on both spins.
There is only one
route for this
The probability
of a total of 4 is
found by
multiplying 0∙3
3
by 0∙3
So this is 0∙09
(b) What is the probability that the total score is 4?
Complete this probability tree for throwing a
fair red dice and a fair blue dice
Red Dice
1
6
Answer
1
6
Blue Dice
Six
Six
5
6
1
6
5
6
Not
Six
Six
Not
Six
5
6
Not
Six
CUMULATIVE FREQUENCY CURVES - 60 students took a test. Question
The graph shows information about their marks
4
60
a) What was the median mark?
55
50
45
b) What was the lowest mark?
40
Cumulative
35
Frequency
30
c) Estimate how many students
scored 12 or less marks
25
20
20
15
d) Estimate the interquartile
range
10
= 4.5
5
2
Answer
5
4
6
15.5
14 11
Marks
8
10
12
14
16
18
20
BOX PLOTS - This box plot shows information about 40 students’
test marks
22
24
26
28
30
32
34
36
38 40
42
44
46
48
50
52
54
Which of these statements are true?
a) The top mark was 54 False - 53
b) The range was 35
False - 30
c) The median was 42
False - 38
True
d) The interquartile range was 12
e) ½ the students scored less than 38 True
Answer