Teaching Calculation

Audit and Test

Richard English, Author

• Publisher: Learning Matters
• Year: 2013
• Print ISBN: 9781446272770, 9781446272763
• eBook ISBN: 9781473904675
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

Can you demonstrate a clear understanding of primary mathematics?
If you are training to be a primary school teacher you need to have, and demonstrate, a clear understanding of primary mathematics. This companion text to the popular Teaching Arithmetic in Primary Schools enables you to audit your knowledge, skills and understanding, making you more aware of the subject and the areas you need to develop further.
It includes:

• self audits on all areas of calculations, supporting trainees to meet the Teachers? Standards
• clear links to classroom practice, linking theory with practice
• advice on next steps for further learning under each chapter

If you?re a trainee primary school teacher, this resource, along with its companion title will provide you with all the guidance and support needed to develop your Primary Maths subject knowledge and teaching skills. 

This book is part of the Transforming Primary QTS Series
This series reflects the new creative way schools are beginning to teach, taking a fresh approach to supporting trainees as they work towards primary QTS. Titles provide fully up to date resources focused on teaching a more integrated and inclusive curriculum, and texts draw out meaningful and explicit cross curricular links.

1 Developing a feel for number

Learning outcomes

This chapter will help you to:

• audit the extent to which you possess the essential knowledge and understanding that underpin calculation;

• develop a better understanding of numbers and the number system;

• identify resources that will help you to develop a feel for number.

Introduction

Before exploring the subject of calculation itself, it is important to ensure that you have a sound understanding of the numbers you will be working with and the fascinating ways in which they relate to one another. The aim of this chapter is therefore to establish the extent to which you possess a good ‘feel for number’ because this is an essential prerequisite for your own competence with calculation and is something that you should be seeking to develop amongst the children you teach.

After attempting the questions in each section below, check your responses with those that can be found at the end of this chapter.

Section 1: understanding numbers – multiples, factors and primes

1. Write down the first few multiples of 5. What do you notice about them? Write down anything at all that you think is of interest. How do they relate to the multiples of 10?

2. Write down the first few multiples of 3. What do you notice about them? Write down anything at all that you think is of interest. How do the multiples of 3 relate to the multiples of other numbers?

3. Write down the first few multiples of 4. What do you notice about them? Write down anything at all that you think is of interest. How do the multiples of 4 relate to the multiples of other numbers?

4. Write down the first few multiples of 9. What do you notice about them? Write down anything at all that you think is of interest. Are there any interesting patterns in the digits?

5. Write down all the factors of 24.

6. Write down all the factors of 35.

7. When listing factors it is often helpful to look for them in pairs (called ‘factor pairs’). For example when listing the factors of 28 you might start with 1 and 28 (because 1 × 28 = 28), followed by 2 and 14 (because 2 × 14 = 28), and so on. So if factors occur in pairs, does this mean that a number always has an even number of factors? Can you find a number which has an odd number of factors?

8. How can you tell, without carrying out any calculations, that 2, 4, 5, 6, 8 and 10 are not factors of 657?

9. Does 657 have any other factors apart from 1 and 657? Do you know any quick ways of checking?

10. Can you check quickly to see if 85,632 is divisible by 4? Which part of the number do you need to focus on? Why?

11. Is 85,632 divisible by 3? Do you know a quick way of checking?

12. Is 85,632 divisible by 6? Do you know a quick way of checking?

13. Is 547,856 divisible by 8? Do you know a quick way of checking?

14. What is the definition of a prime number? What is the first prime number?

Section 2: understanding calculation

Agree/disagree

Read the following statements and decide whether you agree or disagree with each one.

Use examples to explain your decisions. Then compare your thoughts with the notes provided at the end of the chapter.

15. Subtraction always involves ‘taking away’ – i.e. the physical removal of objects.

16. Division always involves sharing a quantity equally into a number of parts.

17. When you add something to a number, the answer is always going to be more than the original number.

18. When you subtract something from a number, the answer is always going to be less than the original number.

19. When a number is multiplied by something, the answer is always going to be more than the original number.

20. When a number is divided by something, the answer is always going to be less than the original number.

More questions for you to try

21. Without actually calculating 283 × 38 decide which of the following is the correct answer: (a) 10,753; (b) 10,754; (c) 10,755. How do you know?

22. Without actually calculating 137 × 49 decide which of the following is the correct answer: (a) 6712; (b) 6713; (c) 6714. How do you know?

23. Without actually calculating 179 × 23 decide which of the following is the correct answer: (a) 4113; (b) 4115; (c) 4117. How do you know?

24. Why do 258 + 137 and 260 + 135 give the same answer?

25. Why do 376 − 268 and 378 − 270 give the same answer?

26. Why do 144 × 3 and 72 × 6 give the same answer?

27. Why do 450 ÷ 36 and 150 ÷ 12 and 75 ÷ 6 and 25 ÷ 2 all give the same answer?

28. What is the commutative law? Give examples to illustrate your answer.

29. What is the associative law? Give examples to illustrate your answer.

30. What is the distributive law? Give examples to illustrate your answer.

What to do next?

Developing your own subject knowledge

To develop your understanding of numbers and the number system, it is recommended that you consult the following sources:

• Mathematics Explained for Primary Teachers, Chapter 6 (‘Number and place value’) and Chapter 14 (‘Multiples, factors, primes’).

• Primary Mathematics: Teaching Theory and Practice, Chapter 7 (‘Number’).

• Teaching Arithmetic in Primary Schools, pages 36–37 (‘Understanding the number system’).

• Understanding and Teaching Primary Mathematics, Chapter 4 (‘Counting and understanding number’).

Also find out about other special sets of numbers that feature in the mathematics curriculum, such as square numbers, triangular numbers, the Fibonacci sequence, Pascal’s triangle, the golden ratio, and so on. Investigate some of the patterns and relationships that exist within these numbers. For example, have a look at the differences between consecutive square numbers. What do you notice? Add any two consecutive triangular numbers. What do you notice? Write down the sum of each horizontal row of numbers in Pascal’s triangle. What do you notice? These are just a few examples of the fascinating patterns and relationships that can be explored within our number system.

Having an awareness of them, or at least possessing the capacity to investigate them, will contribute to your overall ‘feel for number’ and therefore your ability to calculate effectively.

To find out more about the commutative, associative and distributive laws, it is recommended that you read the following:

• Primary Mathematics: Knowledge and Understanding, pages 18–19 (‘The laws of arithmetic’).

• Teaching Arithmetic in Primary Schools, pages 33–36 (‘The laws of arithmetic’).

Developing your knowledge of the curriculum

Read the National Curriculum programmes of study for mathematics and identify those aspects which focus on children’s understanding of the number system. When you have identified these aspects, discuss them with staff in your placement schools and find out how teachers address them. Also have a look at the commercial schemes, resource books, software and other materials that are used by schools to support these aspects of the mathematics curriculum.

Recommended reading

Cotton, T. (2010) Understanding and Teaching Primary Mathematics. Harlow: Pearson Education.

English, R. (2013) Teaching Arithmetic in Primary Schools. London: Learning Matters/SAGE.

Haylock, D. (2010) Mathematics Explained for Primary Teachers (4th edn). London: SAGE.

Mooney, C., Briggs, M., Fletcher, M., Hansen, A. and McCullouch, J. (2012) Primary Mathematics: Teaching Theory and Practice (6th edn). London: Learning Matters/SAGE.

Mooney, C., Ferrie, L., Fox, S., Hansen, A. and Wrathmell, R. (2012) Primary Mathematics: Knowledge and Understanding (6th edn). London: Learning Matters/SAGE.

Answers

Section 1: understanding numbers – multiples, factors and primes

1. Here are some of the things you may have noted about the multiples of 5:

• The final digit is always 5 or 0.

• The final digits alternate between being 5 and 0.

• They alternate between being odd and even.

• The even multiples of 5 are multiples of 10.

2. Here are some of the things you may have noted about the multiples of 3:

• They alternate between being odd and even.

• The even multiples of 3 are multiples of 6.

• Some of the multiples of 3 are multiples of 9 (every 3rd one).

3. Here are some of the things you may have noted about the multiples of 4:

• They are all even.

• Alternate multiples of 4 are multiples of 8.

4. Here are some of the things you may have noted about the multiples of 9:

• They alternate between being odd and even.

• The 10s digit increases by 1 as the 1s digit decreases by 1.

• The digit sum is always 9. For example 27 (2 + 7 = 9), 36 (3 + 6 = 9), 99 (9 + 9 = 18 and 1 + 8 = 9)

• They are all multiples of 3.

5. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

6. The factors of 35 are 1, 5, 7 and 35.

7. Not all ...

Table of contents

• Front Cover
• Halftitle
• Title
• Copyright
• Contents
• Introduction
• 1 Developing a feel for number
• 2 Mental calculation and the recall of number facts
• 3 The development of penciland-paper calculation
• 4 Traditional pencil-and-paper calculation
• 5 Calculating with fractions, decimals, percentages and ratios
• 6 Calculation using technology
• Appendix 1
• Appendix 2
• Appendix 3: Final summary audit/test: 100 questions for you to try
• Appendix 4: Final summary audit/test: answers