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PLT Skills
LESSON OBJECTIVES
Always aim
high!
We are learning to:
- Finding connections between different words. (Which
PLT skills?)
- Accurately solve equations, one linear and one nonlinear, with graphs. (Grades A/A*)
Where are we in
our journey?
AUTHOR
www.mistrymaths.co.uk
Real life
cross/curricular
links?
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PLT Skills
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BRAIN IN GEAR
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EXAMPLE
DITDIONA can be rearranged to make ADDITION
TASK
Work out the following Mathematical anagrams:
UCIADRQAT
Quadratic
SADIUR
Radius
NINETIORDESCT
Intersection
EXTENSION
Develop your own Mathematical anagrams as above as a
creative thinker.
Creative
Thinker
PLT Skills
Effective
Participator
TASK
1) Find the area of:
7cm
8cm
Independent
Enquirer
Reflective
Learner
STARTER
standard form
-4
= 2.3 x 10
86°
130°
115°
Pentagon
adds to =
(
540°
x = 540°- 140°- 130°- 115°- 86°
x = 69°
4x - 6
x + 23
2x + 4
3x + 9 + 2x + 4+ 4x - 6 + x + 23 = 360
= 360
10x + 30
10x
5) Value of:
Reciprocal
3x + 9
- 30
12 x 8
Area of a triangle =
2
Area of a triangle = 48cm2
140°
Which ones are you using?
2) Write 0.00023 in 3)
12cm
x
Team
Worker
Work out x:
5cm
4) Work out x:
Self
Manager
9
-
3
2
EXTENSION
Power
Root
1
9
)3
1
33
1
=
27
=
9m
÷ 10
x
= 33°
6m
x
Work out x:
x
6
x
x
= 330°
÷ 10
7m
2m
- 30
11
2
11
x6
=
2
=
= 33m
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
Quadratic
Linear
EXAMPLE 1
1) Find the approximate solutions to the pair of equations y = x2 + x – 2 and y = 2x + 3
by graphical means.
Set up a table for the quadratic and linear
equations (get the x values from the x-axis)
Table for quadratic
x
y = x2 + x – 2
0 -2 -2 0 4 10 18
10 4
Substitute in the x values and work out y
222222
(4)
(1)
(4)
(1)
----222-2- 2
(-4)
(-4)
(0)
(0)
(-2)
(-2)
(3)
(3)
(-3)
(-3)
(2)
(2)
(-1)++++++
(-1)
12
19
16
0
4+-+
=====16
--313
2404----2
49
1
1222
-+
Table for linear y = 2x + 3
-5 -3 -1 1
3 5 7 9 11
Substitute in the x values and work out y
2(-3)
2(2)
2(-1)
2(-4)++33
2(3)
2(4)
2(-2)
2(0)
2(1)
-6
4
-2
-8++++3333
0
8
-4
2
== 6
Approximate solutions → (-1.8,-0.6) (2.8,8.6)
x
x
x
x
x
x
x
x
x
x
x
x x
x
x
x
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
1) Find the approximate solutions to the pair of equations y = x2 + 3x – 2 and y = x by
graphical means.
Table for quadratic
2
-2
-4 -4
y = x2 + 3x – 2
-2
2
8
x
16 26
x
Table for linear y = x
-4
-3
-2
-1
x x
x
x
0
1
2
3
4
Approximate solutions → (0.7,0.7)(-2.7,-2.7)
x
x
x
x x
x
x
x x
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
2) Find the approximate solutions to the pair of equations y = x2 - 3x – 6 and y = 2x by
graphical means.
y = x2 - 3x – 6
Table for quadratic
x
22
4
-6
-8
-2
12 34
x
x
Table for linear y = 2x
x
x
-8
-4
0
4
8
12 16
x
x
x
Approximate solutions → (-1,-2) (6,12)
x
x
x
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
3) Find the approximate solutions to the pair of equations y = x2 - 3x + 1 and y = 2x - 1
by graphical means.
Table for quadratic
x x
y = x2 - 3x + 1
x
1 -1 -1 1 5 11 19
x
x x
Table for linear y = 2x - 1
x
-1 1
3
5 7
9 11
x x
x x x
Approximate solutions → (0.5,0) (4.6,8.2)
x
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
4) Find the approximate solutions to the pair of equations y = x2 - 3 and y = x + 3 by
graphical means.
Table for quadratic
x
y = x2 - 3
x
x
x
22 13 6 1 -2 -3 -2 1 6 13 22
x
x
Table for linear y = x + 3
x
x
-2 -1 0 1
2 3 4
x
5 6 7 8
x
x
x
Approximate solutions → (-2,1)(3,6)
x
x
x
x
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
5) Find the approximate solutions to the pair of equations y = x2 - 3x - 2 and y = 2x - 3
by graphical means.
Table for quadratic
y = x2 - 3x - 2
x
x
38 26 16 6 2 -2 -4 -4 -2 2 8
x
x
Table for linear y = 2x - 3
x
x
x
x
-13 -11 -9 -7 -5 -3 -1 1 3 5 7
x
x
Approximate solutions → (0.2,-2.6)(4.8,6.6)
x
x x
x
x
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Effective
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PLT Skills SOLVING BY GRAPHING
EQUATION OF A CIRCLE
Equation of a circle → x2
+ y2 = r 2
Self
Manager
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Worker
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with centre (0,0)
Radius of a circle
Draw the circles of the equations given below:
2
2
2
2
2
2
x
+
y
=
5
(b)
(a) x + y = 3 Radius of circle
Radius of circle
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PLT Skills SOLVING BY GRAPHING
EQUATION OF A CIRCLE
Equation of a circle → x2
+ y2 = r 2
Self
Manager
Team
Worker
Which ones are you using?
with centre (0,0)
Radius of a circle
Draw the circles of the equations given below:
2
2
2
2
2
2
x
+
y
=
49
(d)
7
Radius = 7
(c) x + y = 16
4
Radius = 4
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 2 (GRADE A)
1) Find the approximate solutions to the pair of equations x2 + y 2 = 25 and y = 1 - x by
graphical means.
Table for linear y = 1 - x
x
7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5
Radius for circle x 2 + y 2 = 25
2
5
Radius = 5
x
x
x
x
x
x
x
x
x
x
x
Approximate solutions → (-3,4)(4,-3)
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 2 (GRADE A)
2) Find the approximate solutions to the pair of equations x2 + y 2 = 4 and y = x + 1 by
graphical means.
Table for linear y = x + 1
x
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Radius for circle x 2 + y 2 = 4
x
2
2
x
Radius = 2
x
x
x
x
Approximate solutions → (-1.8,-0.8)(0.8,1.8)
x
x
x
x
x
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 2 (GRADE A)
3) Find the approximate solutions to the pair of equations x2 + y 2 = 9 and y = x - 1 by
graphical means.
Table for linear y = x - 1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Radius for circle x 2 + y 2 = 9
x
2
3
x
Radius = 3
x
x
x
x
x
x
x
Approximate solutions → (-1.6,-2.6)(2.6,1.6)
x
x
x
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
EXTENSION (GRADE A*)
1) (a) Find the approximate solutions to the pair of equations y = x 2 + 3x - 4 and
y = 5x - 5 by graphical means.
(b) What is special about the intersection of the two graphs?
(c) Show that 5x – 5 = x2 + 3x – 4 can be re-arranged to x2 - 2x + 1 = 0
(d) Factorise and solve x2 – 2x + 1 = 0
(e) Explain how the solution in part (d) relates to the intersection of the graphs.
2) (a) Find the approximate solutions to the pair of equations y = x 2 + 2x + 3 and
y = x - 1 by graphical means.
(b) What is special about the intersection of the two graphs?
(c) Rearrange x – 1 = x2 + 2x + 3 into the form ax 2 + bx + c = 0
(d) Work out the discriminant b2 – 4ac for the quadratic on part (c)
(e) Explain how the discriminant relates to the intersection of the graphs.
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
EXTENSION ANSWERS(GRADE A*)
1) (a) Find the approximate solutions to the pair of equations y = x 2 + 3x - 4 and
y = 5x - 5 by graphical means. (1,0)
There is only one point of intersection.
x2 + 3x – 5x - 4 + 5 = 0
x2 – 2x + 1 = 0
(x - 1 )(x - 1 ) = 0
x–1=0
+1
x
+1
=1
Only one solution as the line is a tangent
to the curve.
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
EXTENSION ANSWERS(GRADE A*)
2) (a) Find the approximate solutions to the pair of equations y = x 2 + 2x + 3 and
y = x - 1 by graphical means. There is no solution.
The graphs do not intersect.
x2 + 2x – x + 3 + 1 = 0
x2 + x + 4 = 0
x2 + x + 4 = 0
a=1 b=1 c=4
2
b - 4ac
= 12 - 4 x 1 x 4
= -15
The discriminant is negative and you cannot find
the square root of a negative number. Therefore,
there are no solutions.
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PLT Skills TRIGONOMETRIC ANGLES Which ones are you using?
MINI-PLENARY 1 – SPOT THE MISTAKE
Find the approximate solutions to the pair of equations y = x2 - 2x - 1 and y = 2x - 2
by graphical means.
Table for quadratic
y = x2 - 2x - 1
x
34 23 14 8 2 -1 -2 0 2
7
-1
7 16
x
x
x
14
Table for linear y = 2x - 2
x
x x
x
-12 -6 -8 -6 -4 0 0 2 5 6 8
-10
-2
4
x
x
x
x x
Approximate solutions → (0.3,-1.5) (3.7,5.5)
x
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SOLVING BY GRAPHING Which ones are you using?
MINI-PLENARY 2 – SPOT THE MISTAKE
Find the approximate solutions to the pair of equations x2 + y 2 = 4 and y = x + 1 by
graphical means.
Table for linear y = x + 1
x
-5 -4 -3 -2 -1 0 -1 2 3 4 5 6 7
Radius for circle x 2 + y 2 = 4
x
x
2
2
x
Radius = 2
x
x
x
x
x
x
Approximate solutions → (-3.2,-2.2)(2.2,3.2)
(-1.8,-0.8)(0.8,1.8)
x
x
x
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PLT Skills
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Participator
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Reflective
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DISCOVERY
Self
Manager
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Worker
Which ones are you using?
LINK BACK TO OBJECTIVES
- Accurately solve equations, one linear and
one non-linear, with graphs.
What grade
are we
working at?
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
PLENARY ACTIVITY– EXAM QUESTION (GRADE A*)
The quadratic graph has equation y = ax 2+ bx where a and b are integers. Points P(-1,10)
and Q(4,0) lie on this graph. The straight line is y = x. Find the co-ordinate of the point
To find the co-ordinates of R, we
marked R. You must show your working. 16a + 4b = 0
y
a – b = 10 x4 need to find where the equations
of the graphs intersect
Solve the simultaneous
16a
+
4b
=
0
y
=
x
P
equations
y = 2x2 - 8x
y=x
4a
–
4b
=
40
+
(-1,10)
By substitution of x for y,
xy
R
20a
= 40
x = 2x2 - 8x
a
=2
By
substitution,
0 = 2x2 - 9x
x
Q
a – b = 10
By factorisation,
(4,0)
xy
2 – b = 10
0 = x(2x – 9)
x(2x – 9) = 0
Equation for (-1,10) Equation for (4,0)
–b =8
x = 0 or 2x - 9 = 0
b = -8
y = ax2 + bx
y = ax2 + bx
2x = 9
Sub in (-1,10) for x and y
Sub in (4,0) for x and y
y = ax2 + bx
Sub in for a and b
10 = a(-1) 2 + b(-1)
0 = a(4)2 + b(4)
x = 4.5
y = 2x2 - 8x
10 = a - b
0 = 16a + 4b
y = 4.5
a - b = 10
16a + 4b = 0
.
.
.
R(4.5,4.5)
(6 marks)
What have you learnt?
Draw your brain
In your brain, write or draw everything you can remember about
graphing inequalities and shading the region defined by it. It can be a skill
or a reflection, or something else that might be prominent in your brain.
Where are we
in our
journey?
What grade
are we
working at?
Positive
Thinker
Creative
Entrepreneur
Independent
Learner
Reflective
Learner
Responsible
Citizen
Team
Worker
Enterprise Skills SELF ASSESSMENT Which ones are you using?
Plenary Activity
How well do you understand the task?
.
I don’t
understand
I nearly
understand
www.mistrymaths.co.uk
I fully
understand
Positive
Thinker
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Responsible
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Enterprise Skills SELF ASSESSMENT Which ones are you using?
Plenary Activity
WWW (What Went Well)
EBI (Even Better If)
On your post it
notes…
Think about how you
can improve your
work.
www.mistrymaths.co.uk
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
1) Find the approximate solutions to the pair of equations y = x2 + 3x – 2 and y = x by
graphical means.
Table for quadratic
Table for linear y = x
y = x2 + 3x – 2
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
2) Find the approximate solutions to the pair of equations y = x2 - 3x – 6 and y = 2x by
graphical means.
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Self
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
3) Find the approximate solutions to the pair of equations y = x2 - 3x + 1 and y = 2x - 1
by graphical means.
Creative
Thinker
Effective
Participator
Independent
Enquirer
Reflective
Learner
Self
Manager
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
4)Find the approximate solutions to the pair of equations y = x2 - 3 and y = x + 3 by
graphical means.
Creative
Thinker
Effective
Participator
Independent
Enquirer
Reflective
Learner
Self
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 1 (GRADE A)
5) Find the approximate solutions to the pair of equations y = x2 - 3x - 2 and y = 2x - 3
by graphical means.
Creative
Thinker
Effective
Participator
Independent
Enquirer
Reflective
Learner
Self
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 2 (GRADE A)
1) Find the approximate solutions to the pair of equations x2 + y 2 = 25 and y = 1 - x by
graphical means.
Creative
Thinker
Effective
Participator
Independent
Enquirer
Reflective
Learner
Self
Manager
Team
Worker
PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 2 (GRADE A)
2) Find the approximate solutions to the pair of equations x2 + y 2 = 4 and y = x + 1 by
graphical means.
Creative
Thinker
Effective
Participator
Independent
Enquirer
Reflective
Learner
Self
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
TASK 2 (GRADE A)
3) Find the approximate solutions to the pair of equations x2 + y 2 = 9 and y = x - 1 by
graphical means.
Creative
Thinker
Effective
Participator
Independent
Enquirer
Reflective
Learner
Self
Manager
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
EXTENSION (GRADE A*)
1) (a) Find the approximate solutions to the pair of equations y = x 2 + 3x - 4 and
y = 5x - 5 by graphical means.
(b) What is special about the intersection of the two graphs?
(c) Show that 5x – 5 = x2 + 3x – 4 can be re-arranged to x2 - 2x + 1 = 0
(d) Factorise and solve x2 – 2x + 1 = 0
(e) Explain how the solution in part (d) relates to the intersection of the graphs.
2) (a) Find the approximate solutions to the pair of equations y = x 2 + 2x + 3 and
y = x - 1 by graphical means.
(b) What is special about the intersection of the two graphs?
(c) Rearrange x – 1 = x2 + 2x + 3 into the form ax 2 + bx + c = 0
(d) Work out the discriminant b2 – 4ac for the quadratic on part (c)
(e) Explain how the discriminant relates to the intersection of the graphs.
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Thinker
Effective
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Self
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PLT Skills SOLVING BY GRAPHING Which ones are you using?
PLENARY ACTIVITY– EXAM QUESTION (GRADE A*)
The quadratic graph has equation y = ax2 + bx where a and b are integers. Points P(-1,0)
and Q(4,0) lie on this graph. The straight line is y = x. Find the co-ordinate of the point
marked R. You must show your working.
y
y=x
P
.
.R
.Q
x
(6 marks)