Teaching Algebra to Post-16
GCSE resit students
Steve Gough & Sue Hough
Manchester Metropolitan University
s.j.gough@mmu.ac.uk
s.hough@mmu.ac.uk
Today’s Session:
• A brief background to this project
• Algebra materials for the classroom
• What we have learnt so far
Our previous research:
•
12 years research in the use of Realistic
Mathematics Teaching (RME)
•
Projects at KS3 and KS4
•
Published materials at foundation tier
GCSE – “Making Sense of Maths” series
Post-16 GCSE Resit Project (1)
• Funded by Nuffield
• 70 project and 70 control students
• 3 colleges
Post-16 GCSE Resit Project (2)
• 9 hours intervention on Number and
9 hours on Algebra
• Pre-test, post-test, delayed post-test,
attitude questionnaire, teacher
interviews
• Materials designed and delivered
• On going
Characteristics of Post-16 GCSE
Resit students
Characteristics of Post-16 GCSE
Resit students
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Two groups (mature and school leavers)
Need a grade C
There because they had to be
Fear/dislike of maths. Background of
failure
Just missed a grade C
“Just remind me”
Poor study skills
Algebra - 9 hours to cover:
• Solving linear equations (including x
on both sides)
• Drawing graphs
• Collecting like terms
• Rearranging equations
• Factorising and expanding
• Solving word problems
Consider two lessons:
1. Fish and chips – exemplifies the
use of context
2. Word Problems – exemplifies use of
the bar model
At the Chip Shop
Fish
Sausage
Chips
Peas
Gravy
£2.60
£1.20
£1.10
£0.50
£0.40
At the chip shop
I do the price
of 1 fish and 1
lot of chips,
then times
that by 3.
I do the price of 3
fish, then the
price of 3 lots of
chips and add
them up.
Lunchtime orders
At lunchtime, people sometimes come in with
big orders.
Why do you think this is?
One lunchtime, a full order is fish and chips 3
times, sausage and chips twice, fish and peas
twice, and 2 extra portions of chips
Orders over the phone
1. 3(f+c) + 4(s+p) + 1c + 2p
+ 2(f+p).
2. 5f + 4s + 4c + 8p
Orders over the phone
3(f+c) + 4(s+p) + 1c + 2p + 2(f+p).
Made simpler,
3(f+c) + 4(s+p) + 1c + 2p + 2(f+p) = 3f + 3c +c +2p
+2f + 2p
=5f + 4s + 4c + 8p
Simplify
1. 2(s+c) + 3 (f+p) + 3(f+c) + 2p + 3c + 2s
2. 2(f+c+g) + 2(f+c) + f + 3(s+c+p) + (s+c) +
2(c+g) + 3c
3. 5(s+c+g) + (f+c) +2(f+c+p+g) +3(f+c+g) +2s
+2c
Cancelling orders
Sometimes, people phone through an order, then ring up a bit later
and change it.
On day, Jane looked at Azim’s notepad and saw 3(f+c)
She came back to check it a few minutes later and saw that now on
the notepad was 3(f+c) - (f+c)
What do you think has happened here?
Cancelling orders
The order was 3(f+c) - (f+c)
What should Jane wrap up for the customer?
On another occasion, Jane saw on Azim’s pad
3(f+c) – f + c
Is this the same?
What do you think 5(s+c+g) - 2(s+c) means?
What is the simplified order here?
Wrapping it up
People complain when they collect their orders if different bags
have different things in them.
One order was for 12f + 16c
How could this be bagged so that each bag contains exactly the
same?
Wrapping it up
Try to do the same with these orders:
1. 3s + 3c
2. 4f + 2c + 2p
3. 2f + 4s + 6f
4. 3s + 3f + 6p + 9c
5. 12s +9c
6. 12f +8c +4g
Doing the maths
Remember that when we say ‘3 fish’ we are actually talking
about the cost of 3 fish
Expanded
How to say Factorised How to say
form
it in
form
it in
expanded
factorised
form
form
3f + 3c
2f + 2c + 2p
6f + 3c + 3p
3 fish and 3
chips
3(f + c)
2(f + c + p)
3(3f + 2c)
2f + 4c + 2p
3 lots of fish
and chips (or
fish and
chips 3
times)
Word Problems
Word Problems - Ages
Dad is 37 years older than his son Joel.
Dad is 4 years younger than Mum.
The total of their ages added together is 99.
How old is Mum?
Spend one minute trying to
solve this problem on your
own.
Share your ideas
Word problems
Word problems like the age problem
from the previous slide are known to be
difficult to solve- for people of any age.
We will work on a method which
involves drawing bars which will
make the problems much easier to see.
Comparing quantities
Aksa eats 6 packets of crisps a week
Latifa eats 10 packets a week
Which bar refers to Aksa’s ?
Which bar refers to Latifas?
Say how you know
Comparing quantities
Aksa eats 6 packets of crisps a week
Latifa eats 10 packets a week
Which bar refers to Aksa’s ?
Which bar refers to Latifas?
Say how you know
Comparing quantities
Libby has 4 pets
Zahra has 6 pets
Which is Libby’s bar?
Which is Zahra’s bar?
Comparing quantities
Libby has 4 pets
Zahra has 6 pets
Which is Libby’s bar?
Which is Zahra’s bar?
Comparing quantities
Niyal earns £8 more than Neha a week
Which is Niyal’s bar?
Which is Neha’s bar?
Comparing quantities
Niyal earns £8 more than Neha a week
Which is Niyal’s bar?
Which is Neha’s bar?
Comparing quantities
Satnam has 3 fewer coats than Ansa.
Which is Satnam’s bar?
Which is Ansa’s bar?
Comparing quantities
Satnam has 3 fewer coats than Ansa.
Which is Satnam’s bar?
Which is Ansa’s bar?
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.
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.
Word problems – Pocket money
Jack has £8 more money in his pocket than Felicity.
Together they have £20.
How much money does Felicity have?
Start by drawing two bars one
to represent Jack’s money and
one to represent Felicity’s
money
Copy the bars and use them to find how much
money Jack has.
Word Problems - Ages
Dad is 37 years older than his son Joel.
Dad is 4 years younger than Mum.
The total of their ages added together is 99.
How old is Mum?
Try to solve this problem by
drawing 3 bars.
Share your ideas
Solve it
Describe what you see in the
picture
Solve it
Describe what you see in the
picture
Is the picture drawn to scale?
Solving equations – with x on both sides
5x + 4 = 3x + 12
4x + 15 = 8x + 3
6x + 3 = 3x + 21
5x + 10 = 4x + 12
Difficulties in this sector:
• Time constraints
• Reluctant learners, low self-esteem,
poor attendance.
• “I just missed a C”
Difficulties of using RME in this
sector:
• Time constraints
• “Just remind me of the method”
• Not able to sustain models over year
• Building rapport
Potential impact
• Use of models and contexts provided students
with new insights into mathematics
• Many students continued to use the models
during the delayed post-tests
• Impact on their exam result will be assessed in
August 2015
• These materials could form the basis for a two
year GCSE resit course
• Teachers need support to take on this approach
Thanks and Questions?
• Text
About MEI
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and work with industry to enhance mathematical
skills in the workplace
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