MEI PowerPoint Template · 2019-07-03 · The diagram below shows a circle with diameter AC. The...

@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

http://www.acme-uk.org/media/35168/acme assessment of problem solving report - june 2016 - final.pdf



#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




[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 translation of point A to point B is the vector 42


[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




[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




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 length of the diameter is 50.



[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 claimed that the cost of petrol was the same at each petrol station.

Is she correct?


[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.




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



[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



[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





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


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.


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


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


When 𝑥 = 4, 𝑦 = 8.

Model 2


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


𝑥 =4 − 𝑥2

𝑥2 + 1

[3 marks]


𝑥𝑛+1 =4 − 𝑥𝑛

2

𝑥𝑛2 + 1


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

mailto:[email protected]

https://twitter.com/galmeidamaths?lang=en-gb

https://twitter.com/galmeidamaths?lang=en-gb

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

mailto:[email protected]

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

http://www.acme-uk.org/media/35168/acme%20assessment%20of%20problem%20solving%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




[3 marks]

_________________________________________________________

_________________________________________________________

Answer ___________________________

Foundation/Higher

On the diagram below, point A is a vertex of a square.


The translation of point A to point B is the vector [42]


[4 marks]

Foundation/Higher




[1 mark]

A B B A

B

A B

A

Higher




What can you say, with reasons, about the quadrilaterals ABGH and

BCFG?

[3 marks]

_________________________________________________________

_________________________________________________________

_________________________________________________________

Higher


The length of the diameter is √50.



[5 marks]

_________________________________________________________

_________________________________________________________

_________________________________________________________

_________________________________________________________

_________________________________________________________

_________________________________________________________

Answer ___________________________

Foundation/Higher




Sharon claimed that the cost of petrol was the same at each petrol

station.

Is she correct?


[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.




[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



[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





[5 marks]

_________________________________________________________

_________________________________________________________

_________________________________________________________

_________________________________________________________

_________________________________________________________

Answer: ______________________________________________

______________________________________________

Foundation/Higher


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

].


Circle your answer.

[1 mark]

Higher


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


When 𝑥 = 4, 𝑦 = 8.

Model 2


of x.

When 𝑥 = 3, 𝑦 = 4

Higher


𝑥 =4 − 𝑥2

𝑥2 + 1

[3 marks]


𝑥𝑛+1 =4 − 𝑥𝑛

2

𝑥𝑛2 + 1


Given that 𝑥1 = 0.5, calculate 𝑥2, 𝑥3 and 𝑥4 and hence describe why this

iterative formula does not work.

[3 marks]

_________________________________________________________

_________________________________________________________

_________________________________________________________

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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