Teaching Algebra to Post-16 GCSE resit...

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 • 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 £2.60

Sausage £1.20

Chips £1.10

Peas £0.50

Gravy £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?

What do you think 5(s+c+g) - 2(s+c) means?

What is the simplified order here?

On another occasion, Jane saw on Azim’s pad

3(f+c) – f + c

Is this the same?

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

form

How to say

it in

expanded

form

Factorised

form

How to say

it in

factorised

form

3f + 3c 3 fish and 3

chips

3(f + c) 3 lots of fish

and chips (or

fish and

chips 3

times)

2f + 2c + 2p 2(f + c + p)

6f + 3c + 3p

3(3f + 2c)

2f + 4c + 2p

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

• 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