FollifootCE

Primary School

Calculation Guide

This guide outlines the principles strategies that are used consistently across the school. If you would like further advice or support please do not hesitate to contact your child’s class teacher.

Maths is part of everyday life and it is essential to develop the pupils’ knowledge, skills, understanding and confidence in mathematics. We aspire that through effective teaching and learning of mathematics children will develop positive attitudes towards mathematics, will be well prepared for future learning in mathematics and will ‘master’ their year group expectations. ‘Mastering’ essentially means that children have;

•Fluency e.g. quickly and confidently recall and use age appropriate mathematic facts and vocabulary for example number bonds and multiplication tables.

•Reasoning for example explaining a chosen method or explaining why an answer is correct or incorrect.

•Problem solving in relation to their learning for example being able to use mathematical knowledge and understanding to solve problems in different contexts e.g. using addition to calculate with time, money or other units of measurement.

The diagram below illustrates the stages all learners go through on the road to ‘mastering’ a new concept or skill. We are aiming to move our pupils toward being ‘unconsciously competent’ as learners. Pupils reasoning will be secured through evidencing the links between the operations.

To enable effective learning over time a consistent approach is necessary. The strategies outlined are those we consider the most suitable and appropriate for the majority of the children at Follifoot. These are the methods we will teach and model. They are the methods we will expect the majority of our learners to use. We are fully aware that one single method will not always suit all children and that some will be comfortable with variations with these methods or on occasion a different method.

To support consistency, fluency and progression and a depth of understanding the following principles will be applied;

  • Children will be immersed in practical learning to ensure the principles of mathematical concepts are securely understood using appropriate mathematical learning.
  • Mental and written calculation methods will be taught alongside each other with an emphasis on secure and confident recall and application of key facts and vocabulary.
  • Within taught methods emphasis will be placed on efficiency. If a child can complete a calculation mentally or with jottings, they will not be expected to complete a written algorithm.
  • Whilst no longer part of the statutory curriculum, children should also be taught when and how to use a calculator appropriately.
  • Problem solving is embedded in all maths lessons this gives the children the opportunity to apply their reasoning.

One of the key learning principles at Follifoot is the concrete-pictorial-abstract approach. This principle suggests that there are three steps necessary for pupils to develop understanding of a concept. Reinforcement is achieved by going back and forth between these representations.

Concrete representation

A pupil is first introduced to an idea or a skill by acting it out with real objects. In division, for example, this might be done by separating apples into groups of red ones and green ones or by sharing 12 biscuits amongst 6 children. This is a 'hands on' component using real objects and it is the foundation for conceptual understanding.

Pictorial representation

A pupil has sufficiently understood the hands-on experiences performed and can now relate them to representations, such as a diagram or picture of the problem. In the case of division, this could be the action of circling objects.

Abstract representation

A pupil is now capable of representing problems by using mathematical notation, for example: 12 ÷ 6 = 2

Addition

Practical–

  • Adding using equipment for example unifix cubes, counting bears , cuisinaire rods, straws, Numicon and Dienes.
  • Counting on using fingers.
  • Counting on, on a number line or 100 square.

Mental- Sound and secure knowledge of

  • Number bonds to 10, 100, 1000, 10,000 and 1.
  • Counting in groups.
  • Place value and partitioning e.g. 453 = 400 + 50 + 3.

Written – Written stages will have several steps thereby ensuring fluency. There is an emphasis on deepening understanding especially of place value through the practical, jotting and mental stages before using the more formal written methods.

  • A filled in number line.

4 + 5 = 9

Starting from the largest number counting in 1’s or groups to add the amount.

  • An empty number line placing the largest number first and partitioning.

24 + 16

+10 +6

24 34 40
  • Partitioning without a number line, lined up in columns to prepare for formal column addition.

345 + 567 = 912

300 + 500 = 800

40 + 60 = 100

5 + 7 = 12

Application and mastery

At every stage the pupils will practice application of their skills until they are mastered. Problem solving is a part of every mathematics lesson.

Formal Column Addition:

NB Please note position of carry figures. This is consistent across all classes at Follifoot.

Subtraction

Practical – Taking away objects from a group, counting back on fingers, counting back on a number line or 100 square

Mental - Sound and secure knowledge of

  • Counting back in singles
  • Counting back in groups e.g. 2, 5, 10’s
  • Counting back in groups bridging into previous 10 or 100 e.g. 122, 112, 102, 92
  • Knowledge of the inverse 5 + 6 = 11 so 11 – 5 = 6 and 11 – 6 = 5

Written – Written stages will have several steps thereby ensuring fluency. There is an emphasis on deepening understanding especially of place value through the practical, jotting and mental stages before using the more formal written methods.

  • Find the difference by counting up on a filled in number line
  • Find the difference by counting on in steps on a number line

74 – 28 = 46

+2 +40 +4

28 30 70 74
  • Partitioning using Dienes equipment to ensure the place value is understood and secure. Then moving to the more formal written column method.
  • Partitioning – Partition the amount being subtracted.

345 – 281 = 64 e.g. -200, then – 80, then -1

345 – 200 = 145

145 – 80 = 65

65 – 1 = 64

Formal Column Method:

Application and mastery

At every stage the pupils will practice application of their skills until they are mastered. Problem solving is a part of every mathematics lesson, including practical.

Multiplication

Practical – Making groups using apparatus e.g. cubes, counters, Dienes etc. The focus will be on setting out in arrays and relating to repeated addition.

Jottings- Drawing pictures, (arrays) and making jottings.

Mental - Sound and secure knowledge of multiplication tables.Learning tables is a vital part of mathematical knowledge. Fun activities such as chanting tables whilst skipping or bouncing a ball all help to embed this.

Phase one – 2, 5, 10.

Phase two – 3, 4 & 6.

Phase three – 7, 8 and 9.

By Year 4 end pupils should know all their tables up to 12X12 and the equivalent division facts.

  • Children should learn the inverse facts alongside multiplication facts e.g. 6 x 7 = 42 so 42 ÷ 7 = 6 and 42 ÷ 6 = 7
  • Apply multiplication facts to multiples of 10 e.g. 3 x 40 = 160, then 20 x 70 = 1400 then 600 x 40 = 24000
  • Apply multiplication facts to decimals e.g. 6 x 0.5 = 3.0 then 0.2 x 0.3 = 0.06

Written – Partitioning – multiplying any whole number or decimal by a single digit.Written stages will have several steps thereby ensuring fluency. There is an emphasis on deepening understanding especially of place value through the practical, jotting and mental stages before using the more formal written methods.

U x TU e.g. 4 x 16 = 64

4 x10 = 40

4 x 6 = 24

40 + 24 = 64

U x HTU e.g. 123 x 7 = 861

100 x 7 = 700

20 x 7 = 140

3 x 7 = 21

700 + 140 + 21 = 861

Using the above method will ensure that the short multiplication method will flow smoothly, then long multiplication method when place value is secure.

Application and mastery

At every stage the pupils will practice application of their skills until they are mastered. Problem solving is a part of every mathematics lesson, including practical.

Division

Practical – Sharing – taking groups of a given amount from a collection. For example here is a collection of 10 cubes. Also using Deanes. Can you put them into groups of 3. How many groups have been made? Are there any left over? If so how many?

Jottings – To support securing understanding then moving to mental.

Mental- Sound and secure knowledge of multiplication tables and inverse up to 12 x 12.

Ability to apply tables knowledge and inverse to multiples of 10 e.g. 45 ÷ 5 = 9

so 450 ÷ 5 = 90

Reasoning to apply tables knowledge to decimals e.g. 18 ÷ 2 = 9 so 1.8 ÷ 2 = 0.9

Commutative facts.

Written – Using and applying multiplication tables knowledge. Short division using Dienes to secure place value and deepen mastery.Written stages will have several steps thereby ensuring fluency. There is an emphasis on deepening understanding especially of place value through the practical, jotting and mental stages before using the more formal written methods.

Short division:

Long division:

Checking answers with a calculator is also important, as it is a skill we use in adult life. They are used throughout the school.

Application and mastery

At every stage the pupils will practice application of their skills until they are mastered. Problem solving is a part of every mathematics lesson, including practical.

1