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Developing Problem
Solving and
Proportional
Reasoning in GCSE
Resit
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/
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)
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 industry to enhance mathematical
skills in the workplace
• We also pioneer the development of innovative
teaching and learning resources
