Developing Problem

Solving and

Proportional

Reasoning in GCSE

Resit

s.hough@mmu.ac.uk

s.j.gough@mmu.ac.uk

MEI conference 2017

Starter Problems

Answer the problems using a

strategy that you would expect

your GCSE students to use

Session outline

The latest GCSE frameworks place a greater emphasis on

assessing students' ability to problem solve and to answer

questions in proportional reasoning. In this session we look

at how the bar model can provide students with a

unifying strategy for tackling many GCSE questions. This

can be particularly useful to Foundation level GCSE

students and those re-sitting GCSE, many of whom can

find it difficult to know where to start on these type of

problems. We also consider the similarities and differences

to the bar modelling approach originating in The

Netherlands and in Singapore.

Content of 9-1GCSE • Entire body of content outlined by the Department for

Education’s Mathematics GCSE subject content and assessment objectives document, published in November 2013

• There is more content at both Foundation and Higher Tiers

• Content domains: F H

• Number 25% 15%

• Algebra 20% 30%

• Ratio, proportion, rates of change 25% 20%

• Geometry 15% 20%

• Probability and statistics 15% 15%

Our background

• Maths teachers, teacher educators

• Researching trials of the Dutch approach to teaching

mathematics known and RME since 2002

• Most recent trial used Nuffield funding to investigate a

small scale Dutch based intervention with Post 16 GCSE

resit classes

Nuffield Funded Project at MMU

• Tutors from MMU worked with teachers in 3 Post

16 colleges in the North West

• Two modules : Number (12 hours)

Algebra ( 9 hours)

• Materials based on the Dutch based Realistic

Maths Education approach to learning

mathematics (called RME)

Nuffield Report –Impact on students

• “…in the number assessment, the intervention group

improved to a greater extent than did the control group,

between pre-test and post-test”

• “…for algebra…the intervention group performed at a

lower level at both pre-test and post-test, but at a higher

level in the delayed test”

• “Of the 49 students in the intervention group who took

the Number post-test, 36 used an RME approach at

least once”

• In their June GCSE resit, project students did slightly

better than their comparable control students

Nuffield Report –Impact on teachers

• “…teachers described having adopted some of

the modelling approaches as part of their

general practice”

• “…teachers commented on the value of the

models in helping students to solve problems”

• “…teachers pointed to the importance of

embedding the approach in their usual teaching”

Progress made from pre to post

test • Representing questions on a bar enabled

students to make some progress towards finding

a correct solution

• The bar enabled students to use the same

model to answer questions from a variety of so

called different topics

Realistic Mathematics Education

(RME)- the Dutch approach to teaching

maths

• Well researched activities encourage

students to move from informal to formal

representations of maths

• Use of context is sustained throughout

• Use of models to support student

development

• Progress towards formal notions seen as a

long term process

RME adapted for use with Post 16

GCSE resit students in the UK

4 key principles relating to:

• Use of context

• Use of models

• Developing not replacing student approaches

• From informal to formal and back again

A flavour of the materials:

Introducing the bar using the

Dutch approach

Comparison of bar model –

Dutch v. Singapore The Netherlands Singapore

Bar model emerges from representing

a particular context.

E.g. a subway sandwich, a computer

download bar, a ribbon

Bar model tends to be imposed as a

method for solving a range of

problems.

Associated with problems in Number,

Algebra and handling data

Predominantly associated with solving

‘word problems’

Lends itself to use of one bar May involve drawing more than one

bar lined up underneath each other

Labelling of the bar may vary giving

different types of bar

E.g. fraction bar, percentage bar,

double number line, ratio table

Labelling of bar tends to be ‘taught’ as

convention

Sharing Pizza

The cheese and tomato pizza shown in this picture costs

£7.20 to buy in a restaurant.

Two old school friends Kate and Pam decide to share

this pizza one lunchtime. Pam is not as hungry as Kate,

so they cut the pizza into 9 slices, Kate has 5 slices and

Pam has 4.

Sharing Pizza

a) Draw a picture to show how to cut up the pizza, so

that Kate has 5 slices and Pam has 4

b) When they get the bill Kate and Pam decide to pay

for exactly what they have eaten. How much should

each person pay?

Sharing pizza

a) If you and the

person next to

you were to

share one of

the pizzas on

this menu,

which one

would you

choose?

b) Imagine that you were to have twice as much pizza

as the person next to you. Draw a picture to show this

and work out how much you should both pay

Sharing the bill

When Kate goes to the restaurant with her friends, it is

always Kate that works out how to divide the bill. She

often does her calculations on a napkin.

Describe what this napkin tells you about their trip to

the restaurant.

Now do Worksheet N3a

Buying Skirting Board

DIY stores sell skirting board in the same way dress

making shops sell ribbon.

The skirting board above costs £12.80 for 2400mm.

Draw a bar to show this information.

Use your bar to work out the cost of 1500mm of skirting

board.

The Download Bar

This is an image of a download bar on a computer.

1) When did you last see something like this?

2) What information is there in the window?

3) How long has the program been loading so far?

4) How many megabytes (MB) are there left to load?

Can this be true?

How can this be true?

My waist is size 80

You need to sort that out.

Mine’s only 44

Extending the Number Line

Unlike many countries, in the UK we still use two different units

for measuring length. Both of these can be seen on a standard

tape measure.

The tape shows that 6 inches is roughly the same as 15 cm.

What do you think 12 inches is in centimetres?

The Number bar

Daniel drew the number bar above to represent the tape measure

1. Make a copy of this bar and mark four other points on it that

you know in both centimetres and inches.

2. Mark is 5 ft and 3 inches. Alexandra is 1 metre and 72

centimetres. Use the 30cm to 12 inches rule of thumb to

decide who do you think is the taller of the two?

Ratio Tables

A problem when you extend the number line is that it becomes

rather difficult to fit on a page. Some people use a ratio table as a

more flexible version of a number line. The idea is that you can

now fill in other values.

Look at the example above of students having partly filled in a ratio

table for the ruler. Can you see where the numbers in the 3rd column

have come from?

Ratio Tables

The following ratio table was used to work out what 5ft 3 inches

is in centimetres.

1. Give two other combinations of cm and ins you could find

from the table

2. From the ratio table above, write down answers to (i) 225 cm

in inches and (ii) 21 inches in cm

Recipes

Helen decides to use the ratio table

below to work out the ingredients

needed for different numbers of

pancakes.

Pancakes

Makes 8 pancakes

Ingredients

125 g plain flour

1 medium size egg, beaten

300 ml milk

a little oil for frying

Copy the table and fill in four more

columns showing the ingredients

needed for different numbers of

pancakes

Recipes

There are 28 people in Helen’s college group and she wants to

make pancakes for all of them.

1. Decide how much of each ingredient she will need.

2. As part of her Food

Technology course, Helen plans

to make 100 pancakes on

Tuesday 4th March.

a) Use a ratio table to work out

how much milk she will need.

b) If you were shopping for

Helen, how much milk would

you buy?

Comparison of bar model –

Dutch v. Singapore The Netherlands Singapore

Bar model emerges from representing

a particular context.

E.g. a subway sandwich, a computer

download bar, a ribbon

Bar model tends to be imposed as a

method for solving a range of

problems.

Associated with problems in Number,

Algebra and handling data

Predominantly associated with solving

‘word problems’

Lends itself to use of one bar May involve drawing more than one

bar lined up underneath each other

Labelling of the bar may vary giving

different types of bar

E.g. fraction bar, percentage bar,

double number line, ratio table

Labelling of bar tends to be ‘taught’ as

convention

A flavour of the materials:

Introducing the bar using the

Singapore approach

Comparing quantities – Crisp packets

Aksa eats 6 packets of crisps a week

Latifa eats 10 packets a week

Which bar refers to Aksa ?

Which bar refers to Latifa?

Say how you know

Word problems – the swimming party

There were 25 children at Lola’s swimming party.

There were 13 more girls than boys at the party.

How many girls and how many boys at the party?

Look carefully at the

bars drawn and say how

they represent the

information in the

question

Where’s 25?

Where’s the 13

Where’s Lola?

Where’s the boys

Copy the bars and use them to find how many

girls and how many boys at the party.

Answer the GCSE

questions using the

bar model

Website link:

• Please see below the link for the Making Sense

of Maths at GCSE resit materials:

• http://www2.mmu.ac.uk/secondary/msm-at-post-

16-gcse/

Questions s.hough@mmu.ac.uk

s.j.gough@mmu.ac.uk

Starting the ratio table

Menu Vocabulary Opinion 1

Qu1 Opinion

1

Qu2 Opinion

1

Qu1 Opinion

2

Qu2 Opinion

2

Qu1 Answer

Qu2 Answer

Back Forward Cont/d Idea Opinion 2 Answer Opinion 1 Opinion 2 Answer Q 1 Q 2

For the next questions, draw your ratio table and fill in four other sets of

values that you know to be true.

2.

1) A hare travelling at top speed

can run 260 metres in 12 seconds.

What else do you know? (30 secs)

2) 8 metres of ribbon costs £4.80

What else do you know? (30 metre

3) 5 miles is roughly 8 km. What

else do you know? (120 miles)

4) To make pink paint Joel uses a

ratio of 4 parts red to 10 parts white.

What else do you know? (30 RED)

5) A printing machine can produce

24 copies in 30 seconds. What else

do you know? (95 SECS)

6) A multipack of crisps contains 15

bags. What else do you know? (12

MULTIPACKS)

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