@MEIConference #MEIConf2019
#MEIConf2019
Using Reasoning to Embed and
Improve Problem Solving
Godfrey Almeida
Harris Federation Mathematics Consultant
Level 3 Lead London Thames Maths Hub
NCETM PD Lead Level 2 & 3
@GAlmeidaMaths
#MEIConf2019
Aims: Understand why the problems were written
Try the problems and discuss why, when and
how they might be useful
Look at common strategies that might help
students
Look at delivery of questions / lessons
#MEIConf2019
Why? New curriculum
Run up to the exam
Unfamiliar questions
Building resilience
Explore multiple methods/representation
Subject knowledge for teachers
Improve questioning
#MEIConf2019
ACME 2016: Problem solving in mathematics: realising the vision through better assessment
#MEIConf2019
Pedagogy How would you deliver them? / How would you
get students to answer them?
When are they appropriate?
What questions would you ask?
Whole department CPD?
#MEIConf2019
Here are two different charts.
Is it possible that they are representing the same data?
Explain your answer.
[3 marks]
_______________________________________________________________
_______________________________________________________________
Answer ___________________________
There might not be any numbers, but there is something important about the size of each of
the parts of the pie and the bars.
Foundation/Higher
#MEIConf2019
Here are two different charts.
Is it possible that they are representing the same data?
Explain your answer.
[3 marks]
Here the charts represent data where the blue and red sections are the same size and
the green section is half of them. As there is no scale, it is possible that they are
representing the same data. However, they might not as the pie chart could represent
10, 10, 5 and the bar chart could represent 20, 20, 10.
Answer Yes
Foundation/Higher
#MEIConf2019
On the diagram below, point A is a vertex of a square.
The diagonally opposite vertex from A is B.
The translation of point A to point B is the vector 42
What are the coordinates of the other two vertices?
[4 marks]
Draw in point B! Find the centre of the square.
Now rotate the vector from A to the centre 90o in either
direction.
Foundation/Higher
#MEIConf2019
On the diagram below, points A and B are the opposite vertices of a square.
The diagonally opposite vertex from A is B.
The translation of point A to point B is the vector 42
What are the coordinates of the other two vertices?
[4 marks]
The point B is
(5, 4) and the centre
of the square is at
(3, 3).
𝐴𝐵 is the vector 42
𝐴𝐶 is the vector 21
The other two points
are on the
perpendicular from
AC.
This leads to one of
the other corners
being on the vector 1
−2 from C, (4, 1).
The final corner is on
the opposite vector,
i.e. −12
from C,
(2 5).
There are other
ways… find the right
one for youl.
Foundation/Higher
#MEIConf2019
Event A and Event B are mutually exclusive.
Which of these Venn diagrams shows events A and B?
Circle the correct answer.
[1 mark]
A
B
B A
B
A
B
A
Can two mutually exclusive events happen
at the same time?
Foundation/Higher
#MEIConf2019
Event A and Event B are mutually exclusive.
Which of these Venn diagrams shows events A and B?
Circle the correct answer.
[1 mark]
A
B
B A
B
A
B
A
Mutually exclusive events cannot
happen at the same time, so the
circles/sets will not overlap. Top
right!
Foundation/Higher
#MEIConf2019
ABCD and EFGH are straight lines.
ABCD and EFGH are parallel
The lines AH, CF and DE are all parallel
What can you say, with reasons, about the quadrilaterals ABGH and BCFG?
[3 marks]
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
Look at the shapes ABGH and BCFG.
What has to be true if they are congruent?
Higher
Look at the shapes ABGH and BCFG.
What has to be true if they are similar?
#MEIConf2019
ABCD and EFGH are straight lines.
ABCD and EFGH are parallel
The lines AH, CF and DE are all parallel
What can you say, with reasons, about the quadrilaterals ABGH and BCFG?
[1 mark]
There are a couple of misconceptions here. The first is that they are congruent. They are
not. The lines ABC and FGH are parallel and so are AH and CF. This means that AH and
CF are equal in length. However, AB and FG are not necessarily the same length.
The second is that they are similar. They are not. As stated previously, AB and FG are
not the same length, nor is their length stated so there is no evidence here to say they
are similar.
The only thing you can say is that they are both trapezia and that their angles are the
same.
Higher
#MEIConf2019
The diagram below shows a circle with diameter AC.
The length of the diameter is 50.
Do the points B and D lie on the circle?
You must justify your answer.
[5 marks]
_______________________________________________________________
_______________________________________________________________
Answer ___________________________
Draw in AC. What have you created?
What do you know about the angle at the circumference in these shapes?
What mathematics would verify that the
angle is what you say it is?
Higher
#MEIConf2019
The diagram below shows a circle with diameter AC.
The length of the diameter is 50.
Do the points B and D lie on the circle?
You must justify your answer.
[5 marks]
𝐴𝐶2 = 72 + 12 = 50
𝐴𝐶 = 50
And 𝐴𝐶2 = 52 + 52 = 50
𝐴𝐶 = 50
Answer Yes
As AC is the diameter, then
if B and D are on the
circumference then two right
angled triangles should be
formed.
Using Pythagoras’ theorem
with the numbers should
give AC to be the same
value for each triangle. If it
holds true, then all points
are on the circumference.
Higher
#MEIConf2019
Sharon filled up her car with petrol at two different filling stations.
At the first station, she bought 22 litres of petrol and paid £27.06.
At the second station, she bought 18 litres of petrol and paid £21.78.
Sharon claimed that the cost of petrol was the same at each petrol station.
Is she correct?
Show working to justify your answer.
[3 marks]
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
Answer ___________________________
How can you work out the cost per litre?
Don’t forget to conclude!
Foundation/Higher
#MEIConf2019
Sharon filled up her car with petrol at two different filling stations.
At the first station, she bought 22 litres of petrol and paid £27.06.
At the second station, she bought 18 litres of petrol and paid £21.78.
Sharon claimed that the cost of petrol was the same at each petrol station.
Is she correct?
Show working to justify your answer.
[3 marks]
Firstly you must assume that the cost of petrol is proportional to the amount of petrol
you buy. This allows you to scale the costs to calculate the cost per litre at each petrol
station.
Station 1 = 27.06 ÷ 22 = £1.23
Station 2 = 21.78 ÷ 18 = £1.21
Answer No
Foundation/Higher
#MEIConf2019
a) Complete this table, using the numbers 1 to 12 once each, so that each row has the
same total and each column has the same total.
[2 marks]
b) Zara says “it’s impossible to complete this table, using the numbers 1 to 14 once
each, so that each row has the same total and each column has the same total.”
Explain Zara’s reasoning: ___________________________________________
________________________________________________________________
_____________________________________________________ [2 marks]
Re-read your answer: is your
reasoning convincing?
Try different options – copy this grid a few times. Can you predict what each
row will add up to?
Foundation
#MEIConf2019
a) Complete this table, using the numbers 1 to 12 once each, so that each row has the
same total and each column has the same total.
1 + 2 + 3 + … + 12 = 78, so there is a total of 39 in each row.
[2 marks]
b) Zara says “it’s impossible to complete this table, using the numbers 1 to 14 once
each, so that each row has the same total and each column has the same total.”
Explain Zara’s reasoning: The numbers 1, 2, 3 … 14 add up to 105, which is odd, so I
can’t divide them into two rows with equal totals. [2 marks]
12 2 10 4 5 6
1 11 3 9 8 7
Foundation
#MEIConf2019
Amira has three coins in her pocket. They are all different from each other.
Rick has three coins in his pocket. They are all the same as each other.
Rick has twice as much money as Amira has.
What are the coins that they each have?
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
Answer: Amira has ______________________________________
Rick has _______________________________________ [4 marks]
Try a possible combination: If Rick has three 20p coins,
what total does Amira have? Can she have this total with
three different coins?
Be systematic: try EACH possible combination that Rick could have.
Foundation/Higher
#MEIConf2019
Amira has three coins in her pocket. They are all different from each other.
Rick has three coins in his pocket. They are all the same as each other.
Rick has twice as much money as Amira has.
What are the coins that they each have?
If Rick has three 1p coins, Amira has 1.5p in total
If Rick has three 2p coins, Amira has 3p in total i.e. three 1p coins
If Rick has three 5p coins, Amira has 7.5p in total
If Rick has three 10p coins, Amira has 15p in total i.e. three 5p coins
If Rick has three 20p coins, Amira has 30p in total i.e three 10p coins or one 20p and two
5p coins
If Rick has three £1 coins, Amira has £1.50 in total i.e. three 50p coins
Answer: Amira has 50p, 20p, 5p (75p in total)
Rick has three 50p coins (£1.50 in total = 75p × 2) [4 marks]
Foundation/Higher
#MEIConf2019
Umair has one each of the following rectangles: a 1cm by 2cm, a 2cm by 3cm, a 3cm by
4cm, a 4cm by 5cm, and a 5cm by 6cm.
Can Umair fit all these inside a 7cm by 10cm rectangle? Explain your reasoning.
Answer: ________________________________________________________
______________________________________________ [4 marks]
Draw diagrams and try some
options.
Re-read your answer: is your reasoning convincing? Would a picture be clearer than words?
Foundation/Higher
#MEIConf2019
Foundation/Higher Umair has one each of the following rectangles: a 1cm by 2cm, a 2cm by 3cm, a 3cm by
4cm, a 4cm by 5cm, and a 5cm by 6cm.
Can Umair fit all these inside a 7cm by 10cm rectangle? Explain your reasoning.
Answer:
The rectangles might fit, because the total area of the small rectangles = 2cm2 + 6cm2 +
12cm2 + 20cm2 + 30cm2 = 70cm2, which is the area of the big rectangle.
But they won’t fit in!
Put in the biggest rectangle first, and however you do so the space left over can’t be filled
with the smaller rectangles – draw these pictures to show this:
[4 marks]
nothing fits here
nothing
fits here
nothing fits here
#MEIConf2019
Which of the following would give a larger increase?
You must show working to support your answer.
[3 marks]
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
Increase 1
An increase of 60%,
followed by
a decrease of 10%
Increase 2
A decrease of 5%,
followed by
an increase of 50%
If no initial amount is given, you can choose your own. Start with
£1000 each time.
Foundation/Higher
#MEIConf2019
Which of the following would give a larger increase?
You must show working to support your answer.
[3 marks]
This is a compound interest question.
You must get the multipliers correct for each part of each increase.
In increase 1, the multipliers are 1.6 and 0.9. As the percentage changes happen one
after the other, the multipliers should be multiplied to give 1.6 x 0.9 = 1.44
Likewise, increase 2 should be 0.95 x 1.5 = 1.425
This means that increase 1 is larger as the overall multiplier is higher than the one in
increase 2.
Increase 1
An increase of 60%,
followed by
a decrease of 10%
Increase 2
A decrease of 5%,
followed by
an increase of 50%
Foundation/Higher
#MEIConf2019
Which of the following would give a larger increase?
You must show working to support your answer.
[3 marks]
This is a compound interest question.
1000 x 1.6 = 1600 and then 1600 x 0.9 = 1440.
Or 1000 + 60% of 1000 = 1000 + 600 = 1600.
Then 1600 – 10% of 1600 = 1600 – 160 = 1440.
1000 x 0.95 = 950 and then 950 x 1.5 = 1425.
Or 1000 – 5% of 1000 = 1000 – half of 100 = 950.
Then 950 + 50% of 950 = 950 + 475 = 1425.
So Increase 1 is better.
Increase 1
An increase of 60%,
followed by
a decrease of 10%
Increase 2
A decrease of 5%,
followed by
an increase of 50%
Foundation/Higher
#MEIConf2019
Jenny wants to book a taxi.
Abi’s Autos charges a pick-up fare of £2 and then 30p per mile.
Crystal’s Cabs charges a pick-up fare of £3 and then 20p per mile.
Which should Jenny choose? Explain your reasoning.
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
Answer: ______________________________________________
______________________________________________ [5 marks]
Make up a journey distance: 5 miles, for example. What does Abi charge? What does Crystal charge? Make up another distance 20 miles, say. What does Abi charge? What does Crystal
charge? What do you notice?
How could you represent the information you’re given, and the charges you’ve now worked out?
You could draw a graph to show the
prices…
Foundation/Higher
#MEIConf2019
Jenny wants to book a taxi.
Abi’s Autos charges a pick-up fare of £2 and then 30p per mile.
Crystal’s Cabs charges a pick-up fare of £3 and then 20p per mile.
Which should Jenny choose? Explain your reasoning.
For a 5 mile journey:
Abi charges £2 + £1.50 = £3.50
Crystal charges £3 + £1.00 = £4.00
For a 20 mile journey:
Abi charges £2 + £6 = £8
Crystal charges £3 + £4 = £7
For a journey of 10 miles:
Abi charges £2 + £3 = £5
Crystal charges £3 + £2 = £5
Answer: So it depends! For a journey longer than 10 miles Jenny should choose
Crystal’s Cabs, and for a journey shorter than 10 miles she should choose Abi’s Autos. For
a 10 mile journey it doesn’t matter.
[5 marks]
Foundation/Higher
#MEIConf2019
Match the graph to the correct equation.
A B C D
𝑦 = 2𝑥 + 1 𝑦 = 2 + 𝑥 𝑦 + 2𝑥 = 1 2𝑦 + 𝑥 = 1
[3 marks]
In the form y = mx + c
what is the gradient and what is the y-intercept
Are the equations in
helpful forms?
Foundation/Higher
#MEIConf2019
Match the graph to the correct equation.
A B C D
𝑦 = 2𝑥 + 1 𝑦 = 2 + 𝑥 𝑦 + 2𝑥 = 1 2𝑦 + 𝑥 = 1
[3 marks]
This is an
exercise in
identifying
gradient and y-
intercept both
graphically and
algebraically.
Not all of the
equations are in
the form 𝑦 = 𝑚𝑥 + 𝑐
They can be
written as
follows:
𝑦 = 2𝑥 + 1
𝑦 = 𝑥 + 2
𝑦 = −2𝑥 + 1
𝑦 = −1
2𝑥 +
1
2
A B
C D
Foundation/Higher
#MEIConf2019
The graph of 𝑦 = 𝑥 − 4 2 − 3 is shown below.
The graph of 𝑦 = 𝑥 − 4 2 − 3 is translated by the vector 1
−2.
Which of these graphs shows the correct translated graph?
Circle your answer.
[1 mark]
Stop worrying about the algebra!
There is a point that is
labelled…
Foundation/Higher
#MEIConf2019
The graph of 𝑦 = 𝑥 − 4 2 − 3 is shown below.
The graph of 𝑦 = 𝑥 − 4 2 − 3 is translated by the vector 1
−2.
Which of these graphs shows the correct translated graph?
Circle your answer.
[1 mark]
Foundation/Higher
This is another
question that
appears to be a
higher question,
but is accessible
to Foundation
students.
#MEIConf2019
Two mathematical models are described as follows:
Sketch a graph of both models on the same set of axes to show that there is only one
pair of values of x and y where the two models are equal.
[4 marks]
Model 1
𝑦 is directly proportional to 𝑥.
When 𝑥 = 4, 𝑦 = 8.
Model 2
𝑦 is inversely proportional to the square
of x.
When 𝑥 = 3, 𝑦 = 4
Think about the general shape of the graph for each model…
Draw them on the same set of axes and
be careful with model 2!
Higher
#MEIConf2019
Two mathematical models are described as follows:
Sketch a graph of both models on the same set of axes to show that there is only one
pair of values of x and y where the two models are equal.
Model 1
𝑦 is directly proportional to 𝑥.
When 𝑥 = 4, 𝑦 = 8.
Model 2
𝑦 is inversely proportional to the square
of x.
When 𝑥 = 3, 𝑦 = 4
The x2 term in the denominator means
that the y values of the reciprocal graph
will always be positive, so it is always
above the x axis.
There is only one point of intersection,
so there must be only one pair of x and y
values where the two models are equal.
Also given away by the x3 in the previous
question.
Higher
#MEIConf2019
Show the equation 𝑥3 + 𝑥2 + 𝑥 − 4 = 0 can be rearranged to give
𝑥 =4 − 𝑥2
𝑥2 + 1
[3 marks]
The iterative formula
𝑥𝑛+1 =4 − 𝑥𝑛
2
𝑥𝑛2 + 1
is used to try and solve the equation 𝑥3 + 𝑥2 + 𝑥 − 4 = 0.
Given that 𝑥1 = 0.5, calculate 𝑥2, 𝑥3 and 𝑥4 and hence describe why this iterative formula
does not work.
[3 marks]
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
You may want to calculate further solutions to gain
an idea of what is happening.
Higher
#MEIConf2019
Show the equation 𝑥3 + 𝑥2 + 𝑥 − 4 = 0 can be rearranged to give
𝑥 =4 − 𝑥2
𝑥2 + 1
[3 marks]
The iterative formula
𝑥𝑛+1 =4 − 𝑥𝑛
2
𝑥𝑛2 + 1
is used to try and solve the equation 𝑥3 + 𝑥2 + 𝑥 − 4 = 0.
Given that 𝑥1 = 0.5, calculate 𝑥2, 𝑥3 and 𝑥4 and hence describe why this iterative formula
does not work.
[3 marks]
𝑥2 = 3, 𝑥3 = −0.5, 𝑥4 = 3
The question only asks for these to be calculated, but it is clear to see if you keep going
that the solutions are oscillating between -0.5 and 3.
As this is happening, it is not possible to find a solution as the solutions do not converge
on a set value.
Higher
#MEIConf2019
Suggested Strategies Multistep – work one line at a time forwards
Multistep – work one line at a time backwards
Try something… – test some numbers or make
some up
Recall of multiple facts / topic areas
Being able to use simple maths for complex
problems should show a deeper level of
understanding.
#MEIConf2019
Some delivery ideas…
#MEIConf2019
What you will hear from the teacher The teacher poses a maths question, and then gives the pupils time to discuss, and / or write their answers (in books, or mini whiteboards, or on the main board). The teacher walks, watches, listens, all the time selectively noticing.
“What is the answer?” But rarely “right” or “wrong”
“How did you work it out?” “Do you agree with that?” “What do you think?” “And you?” “And you?”
And then the teachers supports them all to think more deeply: “Why does the method work?” “What relationships are you applying?” “What generalities or rules can you deduce / state / refine?”.
The aim is to reach a conclusion that the pupils have deduced, not one they’ve been told.
#MEIConf2019
What you will hear from the teacher
“How…?”
“Why…?”
“What if…?”
#MEIConf2019
What you will hear from students
Pupils using correct mathematical vocabulary
and terminology, and to give their answers in
coherent full sentences. The teachers do so
by
modelling this themselves
teaching vocabulary explicitly
celebrating and reinforcing its correct use
noticing and responding to its omission.
#MEIConf2019
THANKS FOR COMING
Email: [email protected]
Twitter: @GAlmeidaMaths
Geogebra: https://www.geogebra.org/galmeida
#MEIConf2019
About MEI Registered charity committed to improving
mathematics education
Independent UK curriculum development body
We offer continuing professional development
courses, provide specialist tuition for students
and work with employers to enhance
mathematical skills in the workplace
We also pioneer the development of innovative
teaching and learning resources
Using Reasoning to
Embed and Improve
Problem Solving
Godfrey Almeida [email protected]
@GAlmeidaMaths
Aims Understand why the problems were written
Try the problems and discuss why, when and how they might be
useful
Look at common strategies that might help students
Look at delivery of questions / lessons
Why? New curriculum – more demand
Run up to the exam – preparing students
Unfamiliar questions – need for students to be trained
Building resilience
Explore multiple methods/representation
Subject knowledge for teachers
Improve questioning
ACME Guidance on
Problem Solving 2016 http://www.acme-
uk.org/media/35168/acme%20assessment%20of%20problem%20solvin
g%20report%20-%20june%202016%20-%20final.pdf
Pedagogy Things to think about while completing each question:
How would you deliver them? / How would you get students to
answer them?
When are they appropriate?
What questions would you ask?
Whole department CPD?
Considering multiple methods to complete each question and links to
other topic areas.
Another thing to consider, is how you will them as accessible as possible
to your students.
Let’s do some maths… Each question has a suggested tier. This is simply suggesting that
students on that tier may be able to access the question rather than
saying it could appear on that tier in an exam.
Foundation/Higher
Here are two different charts.
Is it possible that they are representing the same data?
Explain your answer.
[3 marks]
_________________________________________________________
_________________________________________________________
Answer ___________________________
Foundation/Higher
On the diagram below, point A is a vertex of a square.
The diagonally opposite vertex from A is B.
The translation of point A to point B is the vector [42]
What are the coordinates of the other two vertices?
[4 marks]
Foundation/Higher
Event A and Event B are mutually exclusive.
Which of these Venn diagrams shows events A and B?
Circle the correct answer.
[1 mark]
A B B A
B
A B
A
Higher
ABCD and EFGH are straight lines.
ABCD and EFGH are parallel
The lines AH, CF and DE are all parallel
What can you say, with reasons, about the quadrilaterals ABGH and
BCFG?
[3 marks]
_________________________________________________________
_________________________________________________________
_________________________________________________________
Higher
The diagram below shows a circle with diameter AC.
The length of the diameter is √50.
Do the points B and D lie on the circle?
You must justify your answer.
[5 marks]
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Answer ___________________________
Foundation/Higher
Sharon filled up her car with petrol at two different filling stations.
At the first station, she bought 22 litres of petrol and paid £27.06.
At the second station, she bought 18 litres of petrol and paid £21.78.
Sharon claimed that the cost of petrol was the same at each petrol
station.
Is she correct?
Show working to justify your answer.
[3 marks]
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Answer ___________________________
Foundation
a) Complete this table, using the numbers 1 to 12 once each, so that
each row has the same total and each column has the same total.
[2 marks]
b) Zara says “it’s impossible to complete this table, using the
numbers 1 to 14 once each, so that each row has the same total
and each column has the same total.”
Explain Zara’s reasoning:
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
[2 marks]
Foundation/Higher
Amira has three coins in her pocket. They are all different from each
other.
Rick has three coins in his pocket. They are all the same as each other.
Rick has twice as much money as Amira has.
What are the coins that they each have?
[4 marks]
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Answer: Amira has ______________________________________
Rick has _______________________________________
Foundation/Higher
Umair has one each of the following rectangles: a 1cm by 2cm, a 2cm by
3cm, a 3cm by 4cm, a 4cm by 5cm, and a 5cm by 6cm.
Can Umair fit all these inside a 7cm by 10cm rectangle? Explain your
reasoning.
[4 marks]
Answer:
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
Foundation/Higher
Which of the following would give a larger increase?
You must show working to support your answer.
[3 marks]
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Increase 1
An increase of 60%,
followed by
a decrease of 10%
Increase 2
A decrease of 5%,
followed by
an increase of 50%
Foundation/Higher
Jenny wants to book a taxi.
Abi’s Autos charges a pick-up fare of £2 and then 30p per mile.
Crystal’s Cabs charges a pick-up fare of £3 and then 20p per mile.
Which should Jenny choose? Explain your reasoning.
[5 marks]
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Answer: ______________________________________________
______________________________________________
Foundation/Higher
Match the graph to the correct equation.
A B C D
𝑦 = 2𝑥 + 1 𝑦 = 2 + 𝑥 𝑦 + 2𝑥 = 1 2𝑦 + 𝑥 = 1
[3 marks]
Foundation/Higher
The graph of 𝑦 = (𝑥 − 4)2 − 3 is shown below.
The graph of 𝑦 = (𝑥 − 4)2 − 3 is translated by the vector [1−2
].
Which of these graphs shows the correct translated graph?
Circle your answer.
[1 mark]
Higher
Two mathematical models are described as follows:
Sketch a graph of both models on the same set of axes to show that
there is only one pair of values of x and y where the two models are
equal.
[4 marks]
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Answer ___________________________
Model 1
𝑦 is directly proportional to 𝑥.
When 𝑥 = 4, 𝑦 = 8.
Model 2
𝑦 is inversely proportional to the square
of x.
When 𝑥 = 3, 𝑦 = 4
Higher
Show the equation 𝑥3 + 𝑥2 + 𝑥 − 4 = 0 can be rearranged to give
𝑥 =4 − 𝑥2
𝑥2 + 1
[3 marks]
The iterative formula
𝑥𝑛+1 =4 − 𝑥𝑛
2
𝑥𝑛2 + 1
is used to try and solve the equation 𝑥3 + 𝑥2 + 𝑥 − 4 = 0.
Given that 𝑥1 = 0.5, calculate 𝑥2, 𝑥3 and 𝑥4 and hence describe why this
iterative formula does not work.
[3 marks]
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Suggested Strategies Multistep – work one line at a time forwards
Multistep – work one line at a time backwards
Try something… – test some numbers or make some up
Recall of multiple facts / topic areas
Being able to use simple maths for complex problems should show
a deeper level of understanding.
Delivery Ideas…
What you will hear from
the teacher The teacher poses a maths question, and then gives the pupils time to
discuss, and / or write their answers (in books, or mini whiteboards,
or on the main board). The teacher walks, watches, listens, all the
time selectively noticing.
“What is the answer?” But rarely “right” or “wrong”
“How did you work it out?” “Do you agree with that?” “What do
you think?” “And you?” “And you?”
And then the teachers supports them all to think more deeply:
“Why does the method work?” “What relationships are you
applying?” “What generalities or rules can you deduce /
state / refine?”.
The aim is to reach a conclusion that the pupils have deduced,
not one they’ve been told.
“How…?”
“Why…?”
“What if…?”
What you will hear from
students Pupils using correct mathematical vocabulary and terminology,
and to give their answers in coherent full sentences. The
teachers do so by
modelling this themselves
teaching vocabulary explicitly
celebrating and reinforcing its correct use
noticing and responding to its omission.
Thank you
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