# Planning a mathematics activity

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### Part 1

### Outline Of Activity And The Year Group It's Intended For

For this activity the children are given a worksheet which consists of a colour by number but in order to colour it the children need to work out the answers to the sums which relate to a particular colour. The children can choose to use unifix to help them work out the problems while working with a friend to complete the activity. The children need to know their number facts between 0 and 20 and to have knowledge of addition and subtraction to complete the activity. This also helps children to build upon these skills.

This activity is intended for Year One children, this is because the Primary National Strategy states that year one's should in using and applying mathematics “solve problems involving counting, adding, subtracting, doubling or halving in the context of numbers, measures or money” this relates to the adding and subtraction of the sums within this activity. “Answer a question by selecting and using suitable equipment, display results using tables and pictures” this part relates to the children using the unifix to work out the problems then using the picture to display their workings.

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In counting and understanding number children of this age need to be able to “read and write numerals from 0-20” during the activity the children will need to write the answers to the sums.

In knowing and using number facts they should be able to “derive and recall all pairs of numbers with a total of 10 and addition facts for totals at least 5 and work out the corresponding subtraction facts” this relates to the addition part of the activity and relating the number facts they already know and build upon them.

Last but not least in the calculating they need to “recognise that addition can be done in any order and understand subtraction as ‘take away' and find a ‘difference' by counting up.” (DfES, No Date)

### Learning Objectives

I can use my knowledge of addition and subtraction to solve problems. I can share my thinking with a friend and I can use a model to show my investigation.

### Questioning

- What do we already know about addition?
- How is subtraction different to addition?
- How do we use our knowledge of number facts to help us solve this problem?
- How many ways do you know of making 10?

These questions are important to ask at the beginning of the activity to find out what the children already know and what they remember from previous sessions and by doing this the children can then use this knowledge to carry out and complete the activity successfully.

### Resources

For this activity in order for the children to complete it they will need:

- a worksheet each
- a pencil
- colouring pencils
- Unifix

### Part 2

Identify and explain the key mathematical concepts, knowledge, skills and understanding that are present within the activity

“Pupils should be taught to: understand addition and use related vocabulary; recognise that addition can be done in any order; understand subtraction both as ‘take away' and ‘difference' and use the related vocabulary; recognise that subtraction is the inverse of addition; develop further their understanding of addition and subtraction; understand why the commutative law applies to addition; choose and use addition or subtraction to solve problems in ‘real life'” (Haylock, 2006: 29)

Children in Year One need to learn:

- Compare, ordering, reading and writing numbers from 0 to at least 20
- Using knowledge of place value to record numbers on tracks and lines
- Learning and counting at least 20 objects
- Recognising that when objects are rearranged the number is the same
- Learning numbers names
- Counting forwards and backwards from 0 to 20, then beyond
- Place value
- Early addition and subtraction and related language and symbols including equals (=) sign
- Solving problems involving counting, adding and subtracting
- Explaining methods and reasoning using related vocabulary

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Majority of these skills are included in the teaching of this activity; before, during and after depending on how well the children grasped and understood the activity. For the children to learn these skills certain structures need to be in place in their learning which can be included within the activity, such as the addition and subtraction structures that the children can use to learn addition and subtraction and ways for them to use it when solving problems.

The commutative law of addition is present within this activity as it can show children that addition can be done in any order i.e. 3+4 = 4+3. “The significance of this property is twofold. First it is important to realise that subtraction does not have this commutative property. For example 10-5 is not equal to 5-10. Second it is important to make use of commutativity in addition calculations.” (Haylock, 2006: 33)

The aggregation structure of addition can also be found within this activity because “a situation in which two (or more) quantities are combined into a single quantity and the operation of addition is used to determine the total. For example there are 15 marbles in one circle and 17 in another: ‘how many marbles altogether?'” (Haylock, 2006: 30) Although the activity is not set out in the way presented by the example from Haylock, the children may interpret the sums in this way or they may find it easier if they present their working of the problem using this structure.

The augmentation structure of addition is when a quantity is increased by another amount and addition is required to solve the problem. “This is the addition structure which lies behind the idea of counting on along a number line and which we might use with young children for experiencing simple additions.” (Haylock, 2006: 31) During the activity the children taking part may find it easier for them to use a number line to solve the problems, therefore this addition structure takes place with the activity.

Along with addition structures taking place there are also a certain number of subtraction structures involved in this activity including: Partitioning, reduction and inverse-of-addition.

Partitioning is when a quantity is partitioned off and subtraction is used to work out many or how much is left. “Partitioning is the structure that teachers (and consequently their pupils) most frequently connect with the subtraction symbol.” (Haylock, 2006: 34) This structure will help the children with the subtraction sums from this activity and any further activities involving subtraction.

The reduction structure can also be present as the children may use it to work out their subtraction problems; this is because “it is simply the reverse process of the augmentation structure of addition. It refers to a situation in which a quantity is reduced by some amount and the operation of subtraction is required to find the reduced value.” (Haylock, 2006: 34) If the children need a sum explained then, depending on the particular child, they may find it easier to understand through this structure as they can count back on a number line until they reach the answer. Therefore this structure is present when taking part in this activity.

Finally the inverse-of-addition structure can be involved as children may prefer to add on from the smaller number up to the larger number and from this working out the answer rather than ‘taking away'. “This subtraction structure might be interpreted as an action on the number line. This is a particularly important structure to draw on when doing subtraction calculations by mental and informal strategies.” (Haylock, 2006: 36)

All of these structures help the children to learn and develop their addition and subtraction skills by providing them with the knowledge of ways to work out problems effectively and efficiently.

To work out the problems within the activity mentally the children may use a variety of structures/methods such as:

“Stepping stones - usually a multiple of 10 or 100 used to break down an addition or subtraction into easier steps. For examples, to find what has to be added to 37 to get to 75, the numbers 40and 70 might be used as stepping stones.

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Examples of our workEmpty number line - a number line without a scale, used to support mental and informal additions and subtractions; numbers involved in the calculation can be placed anywhere on the line provided they are in the right order relative to each other.

Partitioning (into hundreds, tens and ones) - breaking a number up into hundreds, tens and ones as an aid to using it in a calculation. For example, 476 when partitioned is 400+70+6.

Friendly numbers - two numbers that are related to each other in a way that makes a calculation particularly easy; for example, 457 - 257. Often a calculation can be made easier by replacing one of the numbers with a more friendly number close to it and then compensating later.” (Haylock, 2008: 53)

Identify and explain mathematical difficulties and/or misconceptions that may occur while the children are engaged in the activity.

“Learning is more effective when common misconceptions are addressed, exposed and discussed in teaching” (Ofsted 1995 cited in Pepperell et al, 2009:29)

There are a lot of misconceptions that children can have when confronting specific mathematic skills and concepts and “it needs skilful teaching, firstly to indentify the error and then to help the child to understand why the method does not work but with the support the problem can be recognised and a correct method identified.” (Pepperell et al, 2009: 30). Examples of misconceptions that a child can face during this activity relating to counting, addition and subtraction are explained below:

A misconception that children might have when completing this activity is if the child cannot take a larger number from a smaller one then doing it the other way around will be the same. However with subtraction this is not the case. In order to correct this misconception it would be appropriate to provide the children with a different sum “in which the mistake is more obvious because of the size of the numbers. For example 34 - 7 = 33 uses the smaller from larger misconception to give an answer which clearly cannot be true.” (Pepperell et al, 2009: 30)

Saying how many are left when some objects are taken away, by counting how many objects are left is another misconception that children may have as they may not be confident about when to stop counting when taking away (subtracting) in answer to the question ‘How many are left?

Counting along and back on a number track to and from a given position can be confusing for some children as they may only be able to begin counting at one

Counting objects set out in different arrangements may lead to a child inaccurately counting objects when rearranged

Begin to recognise small numbers without counting and that the number of objects is not affected by their position which could be difficult for a child who has no consistent recognition of small numbers of objects

Counting objects that are out of reach children may lack the systematic approaches needed to accomplish this skill

Finding one more and one less than a given number may lead to misunderstood meaning of ‘one more' and ‘one less' as the children may not be able to consistently identify the number before or after a given number.

Saying how many there are altogether by counting all the objects when combining groups for addition, and separating a given number of objects into two or more groups and say how many there are in each group. The misconception can be that the child does not relate the combining of groups of objects to addition and/or does not interpret the counting of all of the objects as an answer to the question ‘How many are there altogether?'

Although this activity is for year one's, year 2 children also have misconceptions about counting, addition and subtraction which is useful to have a knowledge of from a year one teacher's view point so that they can try and reduce the amount of misconceptions children have before they progress through school improving their confidence in mathematics. A few examples are:

Counting on and back in ones and tens can lead to mistakes when counting using teen numbers and/or crossing boundaries.

Identifying pairs of numbers that add to twenty and use known number facts to add mentally children can have difficulty in remembering number pairs totalling between ten and twenty, resulting in calculation errors.

Finding a difference by counting up from the smaller to the larger number can mean children count up unreliably; still counting the smaller number to get one too many in the answer.

Recognising subtraction as taking away, finding the difference and complementary addition can be difficult as the children may not relate finding a difference and complementary addition to the operation of subtraction.

Recognising, for example, that subtracting 13 ‘undoes' adding 13 and vice versa, and that this means that since 4 + 13 = 17, we can state the inverse that 17 − 13 = 4. This can make children feel insecure in making links between addition and subtraction and/or recognising inverses

Developing and recognising patterns to help deduce other addition and subtraction facts. Children may struggle to grasp this concept if they do not readily use number patterns to support calculating, for example:

46 − 5 = 41, so 46 − 15 = 31, 46 − 25 = 21” (DfES, No Date)

Discuss how the activity can be both simplified and adapted to extend learning further, clearly identifying and explaining the mathematics addressed

This activity can be made simpler for children who are less mathematically able as for example it is important to take into account that not all children will be at the same stage of learning and it does not matter that they are not as able as other children but it does mean the teacher needs to provide them with work that is easier for them to deal with while improving their self-esteem and confidence instead of lowering it by giving them work which is not appropriate for the level of learning. This can be done in a number of ways, for example by making some of the sums provided within the activity easier by giving them sums they might be familiar with rather than making them struggle. It is important not to make it too easy as every child needs to be challenged in order to progress. In the EYFS it also states; “Find one more or one less than a number from one to ten.” (DfES, 2008) Therefore, sticking to one more or one less answer sums i.e. 10-1 and 4+1 rather than more than one number gap as it will be simpler for the less able children to deal with. These children may also find it easier to use a model to help them work out the problems such as unifix and number lines whereas it may not be suitable for the more able children as they would more than likely be able to work them out mentally without the use of a model, but that does not necessarily go for all children.

On the other hand for the more able children of the group it is important to extend the activity so that they progress further as well. From the Primary National Strategy it states that children need to be implementing multiplication and division in their learning as well as addition and subtraction in year two: “Derive and recall all addition and subtraction facts for each number to at least 10, all pairs with totals to 20 and all pairs of multiples of 10 with totals up to 100; Derive and recall multiplication facts for the 2, 5 and 10 times-tables and the related division facts; recognise multiples of 2, 5 and 10; Use knowledge of number facts and operations to estimate and check answers to calculations.” (DfES, 1997-2009)

Therefore, by giving the children simple multiplication and division sums which have been covered before as well as addition and subtraction the activity is then improving their mathematical development and understanding of skills by building on previous knowledge and extending it.

### References And Bibliography

DfES, (2009) The Primary National Strategy, London: DfES

DfES, (2007), The Early Years Foundation Stage, Nottingham: DfES

DfES (2009) The National Strategies: Errors and Misconceptions with Addition and Subtraction Available at: http://nationalstrategies.standards.dcsf.gov.uk/node/20854?uc=force_uj (accessed: 02/12/09)

DfES (2009) The National Strategies: Learning Objectives in Mathematics for Foundation Stage, Year One and Two, Available at: http://nationalstrategies.standards.dcsf.gov.uk/strands/34759/34266/110210%2C110211%2C110212%2C110213, (accessed: 03/12/09)

Haylock, D, (2006), Mathematics Explained for Primary Teachers, London: Sage

Hopkins, C., Pepperell, S., Pope, S., (2004), Understanding Primary Mathematics, London: David Fulton

Pepperell, S., Hopkins, C., Gifford, S., & Tallant, P, (2009), Mathematics in the Primary School - A Sense of Progression, Abingdon: Routledge