· Web viewAs children progress with their mental maths skills they will use more compact forms. 3) Short multiplication with carry digits, a more compact form In this case, 6x8=48

MATHEMATICS CALCULATION POLICY

Rationale

At Connaught House School we provide all children with mathematical opportunities every day. Through focussed subject teaching, play and cross curricular activities we aim to instil the children with an inquisitive nature through problem solving and mathematical challenges supported by strong conceptual understanding. We have created this Mathematics Calculation Policy to ensure continuity and progression of skills throughout the school. This policy draws upon on the resources produced by the NCETM (National Centre for Excellence in the Teaching of Mathematics). Progression within each area of calculation loosely follows the programme of study in the 2014 National Curriculum. It has been adapted for use within Connaught House School and includes written strategies, pedagogy and visual representations for each operation from early years to year 6.

At CHS we believe that the use of physical objects and every day contexts for mathematics is vital for younger children. As the children progress through the school we move to pictorial representations and then more abstract work.

As adults, it is easy to assume our own hard-won knowledge is self-evident and gloss over the subtleties that can confound young minds. We have used recent pedagogical research to help uncover and thus address some of these hurdles.

Aims

This document is written as a guide for parents and teachers. It aims to:

Ensure that we share a set of common methods and thus reduce confusion for the children.

Build awareness of important concepts in mathematics and how they are developed in young minds.

Enable all involved to communicate their understanding using a shared mathematical language.

Layout Style

One digit per square as a general rule. Can be altered for e.g. mixed numbers, labelling of number lines.

Short date – see time and date formats, in the next section. 2 square margin – done with a ruler.

Written: Jan 2018 Review: Jan 2020 1

MATHEMATICS CALCULATION POLICYUnderstanding of number Place valueChildren begin by knowing that a number can be linked to the size of a group of objects. As children come to work with larger numbers a good understanding of place value is required. We can no longer count out 158 cubes, or subtract 381 by adding or removing counters. We reinforce the idea of place value through partitioning. We also practise this with deines, hundred squares, number lines and measuring scales. This should be thoroughly understood by the child before they go into working with decimals. Writing fractionsFractions are a new way of using digits, arranged in a pair, each number having a separate significance or meaning. The numerator counts the number of parts which are there, but the denominator tells you what type of part they are. Thirds are actually smaller than halves, despite three being bigger than two when counting.We begin with simple pictures and hands on activities such as making pizzas and go on to fraction walls and more abstract applications.

Time & date formatsWe encourage the children to recognise the different formats in which dates can be represented. For example, understanding that 1st January 2018 can also be written 1/1/18 or similar (e.g. 01.01.2018). The children are expected to use both date formats from Form One onwards dependent on the lesson. As the children’s understanding of time progresses they are expected to be able to write and recognise the time in both digital and analogue format.


MATHEMATICS CALCULATION POLICYAddition & SubtractionJuniors (J1-F3)Initially the children are taught to count on and then back. Over time the idea that addition and subtraction are the inverse is introduced and the children are encouraged to use the inverse operation to verify their calculations.

1) Combining two quantitiesTwo sets of objects are combined to form a larger group.

Similarly, a single large group can have a subset removed to show subtraction.

2) Augmentation/diminution of a quantityThere is a single starting amount, which is increased or becomes larger.Seen in counting on with fingers, or moving along the number line from a starting point.Number lines and hundred squares help children to grasp the relative values of numbers, leading to an understanding of place value and the decimal system.



Using a hundred square with counters.

28 + 15 = 43

Alternately, we could move down a row to add on ten, then count on five more.

For subtraction we would simply move in the opposite direction.

In this written method the two numbers have been decomposed into tens and units, which are added or subtracted separately before being recombined. This method is useful for mental arithmetic and can be adapted for multiplication (see later).

3) Direct comparison A written sum (or representation using objects), showing that the things being added are the same as the total. These can be presented as number facts, or bonds, often demonstrated with Cuisenaire Rods, Numicon, or similar equipment.

As children progress through the school there is a shift towards the abstract. However, understanding is frequently reinforced by using real world objects and contexts.


MATHEMATICS CALCULATION POLICY5) Written methodsForms One and Two are introduced to the formal written method of column addition with and without carrying.

Our preference is for the carried digit to be shown below the answer line.

The children are also introduced to column subtraction with and without borrowing (also known as regrouping or exchanging). The digit in the tens column decreases by one and the units column then increases by ten.

The digits showing the exchange should be noticeably smaller than the original digits.

6) Visual fraction additionIn the Juniors the children are taught to shade and write fractions and then progress onto adding simple fractions with contextual pictorial representation. The children will be also introduced to equivalent fractions.

Addition and SubtractionSeniors (F4-6)



As the children continue to progress through the school they work with increasingly larger numbers, up to millions, as well as decimals to three places. The children continue to work with practical equipment and extend written methods into non-decimal number systems such as time and imperial measure.

1) Fractions

Children begin understanding equivalent fractions with simple pictures.

As we progress, we use fewer pictures.

For the final answer, fractions should be cancelled down to their simplest form.

Improper (top heavy) fractions should be changed to mixed numbers (whole numbers & fractions). Whole numbers should be noticeably larger than the digits in the fraction.

This is an example changing mixed numbers to improper fractions.



An example of how to calculate equivalent fractions when adding. As children get older and gain in confidence some of these steps may be omitted from the written calculation.

2) Addition & subtraction with decimalsTo avoid confusion with place value, “trailing zeroes” can be written in to clarify place value.

The trailing zero reassures the child that they have the correct column for each digit. Decimal points must always be in line.

3) Time- Adding and subtracting durations

The first method here is known as the “bunny hop” method. It is quite versatile and should not be underestimated.

This other method, which is similar to formal written addition/ subtraction, can be confusing. It switches between decimal and base 60 and should be used with due caution.


What time is 3h30min after 4:50pm?

MATHEMATICS CALCULATION POLICYMultiplication & division Juniors (J1-F3)We start by using repeated addition as the basis of what multiplication “is”.

Similarly, division is repeated subtraction and brings in ideas of fairness and equal sharing.

We teach that these are inverse (opposite) operations. Multiplication and division can be linked across four number sentences and children should be able to derive all three complementary

facts from a single statement:


MATHEMATICS CALCULATION POLICYIn a similar vein, we note that 3x4=4x3, as modelled by rectangular arrays or areas.

Times tables should be learned by heart and are introduced gradually as children progress through the school. The first tables to be learnt are the 2, 5 and 10 times tables. We then proceed, in order:

3, 4, 6, 7, 8, 9, 11 and 12.Quick mental recall of each multiplication fact (in any order) enables children to perform more confidently at higher levels.

A secure knowledge of place value is required when multiplying by larger numbers. We can illustrate this using a diagram similar to the one below.


MATHEMATICS CALCULATION POLICYAs the children continue through Juniors they use both the grid method and partitioning.

Seniors (F4-F6)1) Decomposed multiplication grid.

In this decomposed grid, children can multiply each single digit together separately before joining them all back up for the final answer.

This makes the process of short and long multiplication explicit and less mysterious.


MATHEMATICS CALCULATION POLICY2) Long multiplication with total rows and side notes

This is an extended version, using side notes and multiplying each digit of the 35 by each digit of the 421 separately. As children progress with their mental maths skills they will use more compact forms.

3) Short multiplication with carry digits, a more compact form

In this case, 6x8=48 results in the 8 being written down in the units column, the 4 carried beneath the line in the tens column and added on to 3x8=24, making 28 in the tens column (write down an 8, carry the 2 to the hundreds).

4) Simple division

These number facts should have been learned from times tables knowledge, or found by skip counting and keeping track using

fingers.


MATHEMATICS CALCULATION POLICY5) ChunkingThis is an explicit method of division, using repeated subtraction. It shows every step in calculating the answer and children may use as many steps as they like to simplify the process.

The dark orange highlighted numbers show the number of “chunks” removed from the original dividend. The green highlighted numbers are not required but help increase accuracy with difficult divisors. As children improve with mental calculations and grow in confidence, fewer steps are required since larger chunks can be taken.

Beneath the chunking method, we have illustrated the same calculation using the bar method.

6) Short division, going into decimal places

In this example, in the thousands column, the 5 is divided by 4, giving 1 above the line and 1 remainder, carried over to the hundreds column and written by the 3.We then divide the 13 hundreds by 4,

giving 3 in the answer line and carrying the 1 remainder to the tens column, then continue on in that manner.The decimal point and extra zeroes were added, and the answer line extended, as the solver realised they were needed. No remainder is left in this example.



Instead of .25, the fraction one quarter could have been written.



ReferencesMaths vocabulary:- https://www.ncetm.org.uk/resources/42990#glossary

Harris, A. (2000) Addition and Subtraction (available from http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit5/A&S.pdf)

Harris, A. (2001) Multiplication and Division' (available from http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit6/M&D.pdf)

Badger Maths Problem Solving Series, Sharon Shapiro, from Badger Publishing

Appendix 1: Mathematical Equipment


https://www.ncetm.org.uk/resources/42990#glossary


Hundred Square: Cuisenaire Rods:

Number line:

Deines:

Numicon:

Appendix 2: Problem solving approaches


MATHEMATICS CALCULATION POLICYFrom F1 on, our children are provided with regular opportunities to apply skills and knowledge to a range of problems. Below we show some of the wide range of strategies children may use to support their work. It is by no means exhaustive, as the first step in solving a problem is deciding upon a strategy, sometimes inventing new or adapting an old one. Children should be encouraged to think as creatively as possible. All mathematicians approach problems from their own perspective and there is not necessarily a single “correct” route to the final answer.

The four stages in the problem solving process are: Understanding the problem. Make a plan. Solve it! Check and reflect.

Juniors (F1-F3)1) Using equipment and acting outChildren should re-create the question posed using practical equipment.

2) Drawing a picture/diagramChildren should draw or make jottings to help visualise the problem.

3) Finding rules and patterns, making predictions and checkingChildren will use their understanding of a pattern to predict what will come next. It is imperative that the children have the opportunity to check their predictions as this may affect their understanding of the pattern.

4) Finding all the possibilities, writing an organised listChildren will learn to be methodical and organised in the way they generate a solution to a problem which has many possible answers. For example, how many triple-scoop ice cream cones could you create using three different flavours?

5) Drawing and using a tableChildren will organise their answers in table form, allowing them to see any missing information or relationships between different aspects of a problem.

6) Guessing and checking (trial and improvement)Children will begin with an educated guess, which they will check against the conditions of the problem. If not correct, they will revise their answer in a sensible way, trying to get closer to the true answer.

Seniors (F4-F6)1) Creating a tree diagramChildren will work methodically down the list, linking each factor until all possible combinations have been identified.


MATHEMATICS CALCULATION POLICY2) Working backwardsChildren will work backwards methodically to fill in the missing information that has not been provided at the beginning of the problem. It often incorporates the use of inverse operations.

3) Using simpler numbersChildren will begin by solving a simpler problem within the same context before applying the rule to a more complex task. This can be done by replacing large numbers with smaller ones or looking at simpler versions of shapes/patterns.For example, if asked to make their own version of a Dobble deck, we would start by designing a deck with only 3 cards, then increase the number of cards and see how the that works before trying again with more and more cards.

4) Open ended problem solvingChildren will be challenged to explore problems that can be answered in a range of ways and will need to select the most appropriate approach.

5) Analysing and investigatingChildren will analyse what they know and identify what they need to know in order to solve the problem.

6) Using logical reasoningChildren recognise that each piece of information is vital to solve the problem and that by putting the pieces together they will be working towards a solution. A step by step approach here is critical.


MATHEMATICS CALCULATION POLICYAppendix 3: Mental maths, tips and tricks.

Using your hands:Counting on and back using fingers, in units and/or tensNine times table trick

Time: When going on journeys, ask children how long they think it will take and what

time they think you will arrive.Money:

When shopping, ask your child to keep a running total (to the nearest pound) and see how accurate they are when you get to the tills.Counting:

When walking to and from school, ask your child to say the house numbers out loud, identifying odds, evens, multiples of 5, 10, etc. and predicting what will come next – this can be tricky if there are many house without visible numbers, but you should try and work with them to identify the pattern(s).Near doublesTimes tables advice:

Anchor pointsSongs/rhymes – see youtube for a wealth of variable examples, BBC

supermovers also very goodFill in a times table grid with confidence colours, focus on the red and build up

to all greenGames which can be played to help maths skills:

Snakes and ladders, Monopoly, Cluedo, Scrabble, Uno, Battleships, Yahtzee, Connect 4, Snap, Rummy, Knock out whist, Shoot the moon.

Mental division: Using fingers, one hand for quotient, one hand for remainder

IT resources:Doodlemathswww.LGfL.net , various maths resources


http://www.LGfL.net/

Documents

· Web viewAs children progress with their mental maths skills they will use more compact forms. 3) Short multiplication with carry digits, a more compact form In this case, 6x8=48