Getting started in

Adult Literacy and Numeracy

DAY 3 – Session Five

Numeracy Handouts

NU1 Numeracy Diary

NU2a Sample Biography: Maths History Line Graph

NU2b Blank Biography: Maths History Line

Graph

NU3. Brick Wall and Open Door

NU4. The Language of Maths

NU5. The Four Rules Relationship

NU6. Estimate, Calculate, Check

NU7. Broken Calculators

NU8. Spot the mistakes

NU9a. Critical Numeracy Exercise

NU9b. Teaching critical maths

NU9c. Football problems

NU9d. Borrowing money

NU9e. Holidays in the sun

NU10. Numeracy Techniques – possible responses

NU11. Numeracy ideas

NU12. Number Square

NU13. The Four Operations

NU14. Suggested Diagrams for Four Operations

NU15. Evaluation

Numeracy Diary

What did you do ? / Where? /

Why?

/ Please leave blank
Calculated ingredients / kitchen / To make some food

Keep a record of all the things that you do that involve numeracy in one week and note down what the project users do too.

Learner Biography: Maths History Line Graph

Confidence in Numeracy / Maths / 10 /
9 /
8 /
7
6 / /
5 /
4 /
3 /
2
1
Age 5 / 10 / 15 / 20 / 25 / 30 / 35 / 40 / 45 / 50 / 55 / 60 / 65 / 70 / 75 / 80
Liked maths at primary school / Not as good as my friends / Hated maths at school / Failed ‘O’ level / Did lots of DIY / Did a course, more DIY / Manage money better / Course - help kids homework / Use maths for home finance / Use numeracy more at work / Another course – better job? / Help grandchildren / Feel happy, confident I can achieve new things – pass this on to my children and grandchildren – teach others maths?

Biography: Maths History Line Graph

Confidence in Numeracy / Maths / 10
9
8
7
6
5
4
3
2
1
Age 5 / 10 / 15 / 20 / 25 / 30 / 35 / 40 / 45 / 50 / 55 / 60 / 65 / 70 / 75 / 80

What factors have presented a brick wall to Numeracy?What factors have helped to open the door to Numeracy

The Language of Maths

We use a vast number of words to represent the four operations.

It is our job to respect and build on the language a student may already have and, if necessary, change their usage of the words to fit in with the students.

 / 
add / multiply
sum / times
and / product
plus / lots of
total / by
greater than / five threes
altogether / (a bracket)
more
increase
 / 
subtract / divide
take away / share
minus / goes into
less than / split into
difference / how many times
from / out of
smaller than / between
decrease / (fraction line)
remove

The Four Rules Relationship

The Four Rules of , ,  and  are very closely related and tutors and students must understand the relationships between them.

 /  repeated addition is  / 
is the / is the
opposite / opposite
of / of
 /  repeated subtraction is  / 

Estimate, Calculate, Check

As a tutor, as your student is developing his/her understanding of the relationship between the four rules, then these three actions of estimating, calculating and checking become very useful to demonstrate the relationship between the four rules.

Addition432 + 275

Estimate400 + 300 = 700 (to the nearest 100)

Calculate432 + 275 = 707 (or could lay out in ‘sum’ format)

Check (this is the part that shows the relationship between the rules)

To check answer you do the opposite of what the question was.

This example was an addition, so we do a subtraction to check if we

are correct.

707  275 = 432(or 707  432 = 275)

because addition is commutative

Subtraction224  108

Estimate200  100 = 100 (to the nearest 100)

Calculate224  108 = 116 (or could lay out in ‘sum’ format)

Check (this is the part that shows the relationship between the rules)

To check answer you do the opposite of what the question was.

This example was a subtraction, so we do an addition to check if we

are correct.

116  108 = 224

Multiplication36  82

Estimate40  80 = 3200 (to the nearest 10 in the question)

Calculate36  82 = 2952 (or you could lay out in ‘sum’ format)

Check (this is the part that shows the relationship between the rules)

To check answer you do the opposite of what the question was.

This example was a multiplication, so we do a division to check if we

are correct.

2952  36 = 82(or 2952  82 = 36)

because multiplication is commutative

Division552  12

Estimate550  10 = 55 (to the nearest 10 in the question)

Calculate552  12 = 46 (or you could lay out in ‘sum’ format)

Check (this is the part that shows the relationship between the rules)

To check answer you do the opposite of what the question was.

This example was a division, so we do a multiplication to check if we

are correct.

46 12 = 552

N.B.Both addition and multiplication are commutative (ie. it doesn’t matter which way you add or multiply the numbers you will still get the same answer) but subtraction and division are not commutative.

Broken Calculators

Eh?? What does this mean? – surely you just throw it away and buy a new one!

Well, actually, the ‘broken calculator’ idea is very useful in developing your students’ understanding and confidence in the four rules and the relationship between them.

Your students are told that their calculators are broken and that certain keys do not function. They then have to solve some numerical problems using only the other keys that are working. The bigger the number of keys that are ‘broken’, the harder the task will be.

There are numerous websites devoted to this kind of activity. Some are listed below but, to find others, just type ‘broken calculator’ into any search engine.

Useful Websites

Broken Calculators

Examples

  1. Students are told that certain functions/numbers are not working (you can decide which) and they must produce all of the numbers up to 20, say, without using any of the ‘broken’ keys.

This could be played as a group game with the fastest student to suggest the answer winning that round. Students could also work in pairs or small groups.

  1. Give the student(s) a target number to reach, say 18, but tell them that the ‘8’ on the calculator is broken so they must find another way of eventually displaying ‘18’ on the screen.

This could be achieved by pressing:9  7  2

or9  2

or72  4

  1. Other examples could be:

Calculate840 using only the 3, 2, 5 and 7 keys

Calculate2340  12 without using the 3 and 1 keys

Calculate147 using only the 1 and 2 keys

Spot The Mistakes

Many times you will be faced with a sheet of work that is totally incorrect. It is not appropriate to send the learner away to do the work again unless some attempt has been made to discover the reason for the errors. Listed below are some of the more common errors that occur. Try to discover the misunderstandings or difficulties that the learner is experiencing, and suggest methods to help them understand the reason for the errors. All calculations below have been done on paper - not as mental calculations.

1.A recipe states that the ingredients required weigh 350g, 75g and 200g, and the question asks for the total weight. The learner has arrived at an answer of 1300g - why?

2.Another learner comes up with an answer of 5125g. Why?

3.If you have £3.00 and spend £1.38, how much do you have left? The answer given is £2.38 - why might this be?

4.You have made 25 minutes of calls with your mobile phone. This costs you 35p per minute. How much would the total bill be for this call? The answer given is 60p - why?

5.If you have 40p in your hand and are given another 38p, how much would you have then? The answer given is 70p although the learner realises this is incorrect, they don’t know where their calculations went wrong. Can you help?

6.You have £4.00 in your pocket, and spend 38p, how much do you have left? A learner answers £2.72 - why?

7.The phone bill last month contained an amount for 36 minutes of extra calls. This month it is for 28 extra minutes. The learner has calculated that this should be 91 minutes in total. What went wrong with the calculation?

8.A learner is asked to write their date of birth. It looks like this 21190085. Why is this incorrect?

9.You have 92p and spend 47p - how much do you have left? The answer given is 55p - why?

10.If an interview is at 10.10am and the journey takes you 55 mins, when should you leave the house? Why would a learner say 9.55am?

Critical Numeracy Exercise

In your small group please work with other group members to carry out the following exercise:

  • What sort of activity could you imagine carrying out with this group to encourage critical numeracy thinking?
  • What sort of object and/or set of statistics could you imagine inspiring this project?
  • What are the obstacles to this approach?
  • What are the benefits to this approach?

Teaching critical maths

From Barnes M., Johnston B. & Yasukawa K. 1995, ‘Critical Numercy’, a poster

presented at the Regional ICMI conference, MonashUniversity, April.

A critical approach to mathematics teaching would encourage students to pose their own questions, arising either from stimulus material provided by the teacher or from the students’ own interests and concerns. Students would be encouraged to ask questions such as:

  • What mathematical questions arise out of this situation?
  • What mathematics is being used, or could be used, in this context?
  • Which groups in the community are affected by the circumstances described?
  • Which groups are likely to benefit from the use of mathematics in this context?
  • Could you look at the questions in a different way? Would this produce different answers?
  • Are there important factors which have been ignored?
  • Is there any information not given here which might help you answer your questions?

Questions for critical numeracy

From Keiko Yasukawa 1995, Notes on Critical Numeracy for

the Graduate Diploma of Adult Basic Education, UTS.

  • Who is the potential audience?

Who are the writers?

What are some of the features which effective and responsible communication ought to entail?

  • What sorts of information are of interest?

For whom? From whom?

What are your materials trying to convey?

What issues of power, equity and social justice might be involved?

  • What sorts of language, maths, texts and other means are employed to convey the information?

Does the use of these limit the audience?

Do they distort the information for some and not for others?

Who gains and loses out of the distortions?

  • What maths is being used explicitly or implicitly?

Is there any other maths that could be relevant?

Football Problems?

You are working with a group of young people who are very interested in football – in fact, that is all they talk about. Unfortunately you are supposed to be trying to improve their numeracy skills – so how can this topic be used to gain their attention?

Try to list below all of the ways in which you can think of linking numeracy to football.

How may this be developed to form some sort of group project?

Borrowing money?

Imagine that you have a learner who needs £1 000 for a holiday that they have booked. They have identified as one of their personal goals that they want to know how to decide the best way to borrow the money, and how much it will cost them.

  • Discuss how you could plan an activity to help your learner discover for themselves all the different agencies that will lend money. Include the Internet, newspapers, leaflets etc.
  • Using the figures below discuss why the two APR rates are so different. Calculate the total money paid back and identify the differences and the reasons for it.Discuss how you can use this activity to help your learner understand APR.

Amount borrowed / Length of Loan / APR / Monthly repayments / Total to be paid back
1 000 / 1 year / 18.8% / 91.37
1 000 / 3 years / 18.8% / 35.83
1 000 / 5 years / 18.8% / 25.04
10 000 / 1 year / 8.6% / 867.73
10 000 / 3 years / 8.6% / 314.65
10 000 / 5 years / 8.6% / 204.10
  • Discuss all the skills that a learner requiresto explore the best method of borrowing £1 000 for themselves.

1

Prices per person: Based on sharing a twin room with half board. / Departures between
01 Jan '08 - 31 Mar '08 &
01 Jan '09 - 31 Mar '09 / Departures between
01 Apr '08 - 30 Jun '08 &
01 Apr '09 - 30 Jun '09 / Departures between
01 Jul '08 - 30 Sep '08 &
01 Jul '09 - 30 Sep '09 / Departures between
01 Oct '08 - 31 Dec '08 &
01 Oct '09 - 31 Dec '09
Single supplement = 30% of the basic holiday (except suites - not available for sole occupancy).
Full Board supplement = an additional £100 per week
Sandcastle Hotel - no single supplement applies.
1
Week / 2
Weeks / Extra
Weeks / 1
Week / 2
Weeks / Extra
Weeks / 1
Week / 2
Weeks / Extra
Weeks / 1
Week / 2
Weeks / Extra
Weeks
Coral Beach Hotel
without air-conditioning / 799 / 949 / 150 / 749 / 869 / 120 / 839 / 999 / 160 / 819 / 979 / 160
Trade Winds Hotel
with air-conditioning / 849 / 1049 / 200 / 799 / 969 / 170 / 889 / 1099 / 210 / 869 / 1079 / 210
Silver Sands Hotel
without air-conditioning / 869 / 1089 / 220 / 819 / 1009 / 190 / 909 / 1139 / 230 / 889 / 1119 / 230
Silver Sands Hotel
with air-conditioning / 889 / 1129 / 240 / 839 / 1049 / 210 / 929 / 1179 / 250 / 909 / 1159 / 250
ParadiseBeach
with air-conditioning / 909 / 1169 / 260 / 859 / 1089 / 230 / 949 / 1219 / 270 / 929 / 1199 / 270
Palm Hotel
with air-conditioning / 929 / 1209 / 280 / 879 / 1129 / 250 / 969 / 1259 / 290 / 949 / 1239 / 290
Sandcastle Hotel
with air-conditioning / 929 / 1209 / 280 / 879 / 1129 / 250 / 969 / 1259 / 290 / 949 / 1239 / 290
Seafront Hotel
with air-conditioning / 949 / 1249 / 300 / 899 / 1169 / 270 / 989 / 1299 / 310 / 969 / 1279 / 310
Golden Beach Hotel
with air-conditioning / 979 / 1309 / 330 / 929 / 1229 / 300 / 1019 / 1359 / 340 / 999 / 1339 / 340

Holidays in the Sun

1

Holidays in the Sun

  1. Use the price list provided to produce at least 2 two part questions for a learner.

e.g. a) How much would it cost for two people to spend 3 weeks half board in the Silver Sands hotel starting on May 15th 2009 if they require air conditioning?

b) What would the same holiday cost for a single person?

c) How much would the same holiday cost if they couple require full board and if they do not want air conditioning?

d) What would be the difference in the total cost for two people staying for two weeks half board in the Coral Beach Hotel starting on October 10th 2008, and the same holiday in the Golden Beach Hotel?

Think carefully about the language that you use in all of the questions.

  1. Team up with another participant and take it in turns to pretend to be the learner or the tutor. As the tutor you have to explain the method of solving the two questions that they have written. As the ‘learner’, you may take as long as you like before understanding the question!
  1. The ‘learner’ should then feedback to the ‘tutor’ on the difficulty of understanding the questions.
  1. Make a list of all the words that have been used instead of add, take away, multiply and divide.
  1. Discuss all the skills required by the learner in order to understand and use this price list. Would it have been easier to understand if it had been laid out differently?

Numeracy Techniques – possible responses

In pairs draw up lists of arguments to support and oppose both of the following statements:

  1. We should encourage adult numeracy learners to use calculators.
  1. We should encourage adult numeracy learners to used multiplication tables.

Support

/

Oppose

1) /
  • Useful for checking answers
  • Number keypads are common in everyday literacy, therefore familiar
  • Useful for complicated calculations
  • Quicker
  • Useful tool for some disabled people
  • May make learner more positive about using calculations
/
  • Need to have a rough estimate of the answer do you know if it is right
  • Only useful if you know how to use them
  • Easy to make mistakes
  • Not useful if you don’t know the four operations: addition, subtraction, multiplication, division
  • Discourages learners from thinking for themselves

2) /
  • Quicker to do multiplication calculations
  • Gives confidence in numeracy
  • Speed in mental calculation
  • Useful for estimation
  • Quicker than adding
/
  • Encourages learning without understanding
  • Difficult for some people to memorise
  • Decontextualises learning
  • Learner may have unpleasant memories of learning multiplication at school

Numeracy ideas

  1. The 9 times table
  • You can use your fingers to work out the 9 times table. Extend your fingers in front of you, and imagine them numbered from 1 to 10, left to right.

To work out 9 x 3, fold finger number 3 down (the finger third from the left). You know have 2 fingers to the left and 7 fingers to the right, therefore 9 x 3 = 27.

To work out 9 x 6, fold down finger number 6 (the thumb of your right hand). This gives you 5 fingers to the left and 4 to the right, therefore 9 x 6 = 54.

This method only works for the 9 times table.

  • In the 9 times table, the answers when added together equal 9, e.g.

18 1 + 8 =9

27 2 + 7 = 9

363 + 6 = 9

454 + 5 = 9

  1. How many days in each month of the year?

Put both hands in front of you and close them into fists – you will notice your knuckles stand out with grooves or depressions between them. Start from the far left knuckle and make it represent January, with the depression immediately to the right of it representing February and the knuckle immediately to the right of this representing March and so on (do not count the knuckles of the thumbs). For every month that falls on a knuckle there are thirty-one days. For every month that falls on a depression there are thirty days (excepting February).

Both the above methods are easier to demonstrate than to explain in writing.

Number square

1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
2 / 4 / 6 / 8 / 10 / 12 / 14 / 16 / 18 / 20
3 / 6 / 9 / 12 / 15 / 18 / 21 / 24 / 27 / 30
4 / 8 / 12 / 16 / 20 / 24 / 28 / 32 / 36 / 40
5 / 10 / 15 / 20 / 25 / 30 / 35 / 40 / 45 / 50
6 / 12 / 18 / 24 / 30 / 36 / 42 / 48 / 54 / 60
7 / 14 / 21 / 28 / 35 / 42 / 49 / 56 / 63 / 70
8 / 16 / 24 / 32 / 40 / 48 / 56 / 64 / 72 / 80
9 / 18 / 27 / 36 / 45 / 54 / 63 / 72 / 81 / 90
10 / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / 100

Some ideas for using a number square:

  • You can read off the answer to any multiplication sum up to 10 x 10.
  • Take some numbers out and get learner to complete the square.
  • Halve the number of facts you need to learn by remembering, for example, that 6 x 2 is the same as 2 x 6.
  • Highlight the multiplication facts you need to work on. Usually they are some of the ones in the bottom right-hand corner of the table.
  • Count aloud across a line to practise a table: 5, 10, 15, 20…
  • Look at factors: for example, how many ways can you make 24?
    Answer: 3 x 8, 2 x 12, 4 x 6, 8 x 3, 6 x 4, 12 x 2.
  • The number square will help learners with division and working with fractions.
  • Use the square to look at equivalent fractions: 1/2 = 3/6 = 5/10.

The Four Operations

When a learner has difficulty understanding the basic concepts of any of the four operations it is very useful to be able to demonstrate it by way of a diagram that the learner can relate to. Draw a simple representative diagram for each problem given below.