# Digging Deeper into … Addition and Subtraction (3rd to 6th class)

## Digging Deeper into … Addition and Subtraction (3rd to 6th class)

Category : Uncategorized

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of addition and subtraction, please check out the following post: Dear Family, your Operation Maths Guide to Addition and Subtraction

Addition and Subtraction is always the first operation’s chapter in Operation Maths 3-6, and it is always a double chapter i.e. it is structured to be covered over 10 days/two school weeks. In Operation Maths 3-5 there is also a second Addition and Subtraction chapter (this time only a single i.e. one week chapter) in the second half of the school year to revise and re-focus on specific strategies that can be used.

### Relationship between addition and subtraction

In contrast to traditional maths schemes, which often have separate chapters for each operation, Operation Maths instead teaches addition and subtraction together, as related concepts. Teaching the operations in this way will encourage the children to begin to recognise the relationships between addition and subtraction, and indeed all the operations. Thus, the initial activities in the Discovery Book, require the children to reflect on their understanding of the concepts and to compare and contrast them.

In particular, the children are enabled to understand addition and subtraction as being the inverse of each other and are encouraged to use the inverse operation to check calculations.

### Looking at the bigger picture

Children can often have tunnel vision (or column vision) regarding addition and subtraction calculations: they “do” the units, then the tens, then the hundreds without really looking at the whole numbers or the processes involved.

One way in which you can encourage the children to look at and understand these operations better is by using a Concrete–pictorial–abstract (CPA) approach. This means the children will be moving from experiences with the familiar base ten concrete materials (e.g. straws, base ten blocks, money, the Operation Maths place value discs, pictured above) to pictorial activities (e.g. where the children draw representations of the numbers using pictures of the concrete materials or use empty number lines, bar models, etc.) and finally to abstract exercises, where the focus is primarily on numbers and/or digits.

When exchanging tens and units or tens and hundreds, reinforce that a ten is also the same as 10 units, and that a hundred is the same as 10 tens and 100 units.
The use of non-canonical arrangements of numbers (e.g. representing 245 as 2H 3T 15U or 1H 14T 5U), as mentioned in Place Value, can also be very useful to children as they develop their ability to visualise the regrouping/renaming process. The Operation Maths Place Value eManipulative, accessible on edcolearning.ie,  is an excellent way to illustrate this and explore the operations in a visual way. Worth noting also, is that the Operation Maths Place Value eManipulative and place value discs provide the only means to concretely or pictorially represent base ten materials to five whole number places (no other interactive tool is available on the internet to do this); a fact which will be of particular value to teachers of 5th and 6th classes who didn’t have a way to concretely/visually represent numbers to ten thousands prior to the inception of Operation Maths.

### Mental strategies are as important as written methods

The traditional, written algorithms for addition and subtraction, i.e. the column methods, are important aspects of these operations. However, in real-life maths, mental calculations are often more relevant than written methods. Also, as mentioned previously, children can often have tunnel vision (or column vision) regarding addition and subtraction calculations; they ‘do’ the units, then the tens, then the hundreds, etc., without really looking at the entire numbers or the processes involved. Therefore, while the column method for addition and subtraction is a main part of this topic, equally important is the development of mental calculation skills, using such strategies as those outlined on this page from Operation Maths 6 (below)

Thus, one of the main purposes of the Addition and Subtraction chapters in Operation Maths is to extend the range of mental calculation strategies the children have and to enable them to apply the strategies to numbers of greater complexity i.e. for the children to become efficient and flexible, as well as accurate. As the same calculation can often be done mentally in many different ways, the children have to develop their decision-making skills so as to be in a position to decide what is the most efficient strategy to use in each situation.

It is worth noting that the page from Operation Maths 6 pictured above serves as a synopsis to remind the children of all the strategies they explored individually in the previous Operation Maths books. That said, if the sixth class children are new to Operation Maths and have never encountered these strategies before, they may need to progress at a much slower pace than those who have been using the programme previously, or who may have encountered these strategies, for example a class who used Number Talks. As mentioned in a previous post, the Operation Maths mental strategies listed below are very similar to, and in some cases identical to, those used in Number Talks (if different terminology from Operation Maths is used in Number Talks, the Number Talks terminology is given in brackets).

• Doubles and near doubles
• Number bonds of 10, 100 and 1,000 (Making tens)
• Friendly or Compatible numbers (benchmark/friendly numbers)
• Partitioning (breaking each number into its place value parts)
• Compensation
• Subtraction as take-away (removal/deducation)
• Constant difference subtraction (see below)

Operation Maths also places particular emphasis on the development of estimation skills for number and introduces and develops specific estimation strategies as the books progress. Again, the emphasis is on the children contrasting and comparing these strategies and choosing the most efficient strategy each time. To find out more about some of the estimation strategies, read this post.

Therefore, ask the children, as often as possible when meeting new calculations, can they do it mentally, and how, so that they become increasingly aware of a range of mental calculation skills and approaches. In this way the children will also be developing their decision-making skills, so as to be in a position to decide the most efficient strategy/approach to use.

### Problem-solving strategies

One of the main aims of Operation Maths 3-6 was to introduce both teachers and pupils to a logical problem solving approach (i.e. RUCSAC) , complemented by specific visual problem solving strategies which develop in complexity as the child progresses through the senior classes.

A key step in the RUCSAC problem-solving approach is the ability to read a word problem meaningfully, and highlight the specific operational language or vocabulary. This is reinforced with activities in the Discovery Book (see below) where the children colour-code the specific phrases and then transfer them to their Operations Vocabulary page towards the end of their Discovery Book for future reference.

You will notice that the problems have no numbers to distract the children, so that they can just focus on the language of the problems and the operations that may be inferred by the context of the story. These type of “numberless word problems” are being used more and more by practitioners in order to deepen children’s understanding of the concepts involved.

Another key step in the RUCSAC approach is the ability to create to show what you know, where the child makes a representation of the word problem in another form. Bar models are ideal for use with operational word problems. Introduced initially in Operation Maths 3, the use of bar models is developed through Operation Maths 3-6 to include bar models suited to other types of word problems.

Empty number lines can also be used to represent addition and subtraction problems (see below). In the senior books, the children will use both strategies to represent word problems and compare and contrast the two strategies. Ultimately, it is hoped that the children will use the strategy that they are most comfortable with. For more information on problem-solving strategies please consult the guide to problem-solving strategies across the scheme in the introduction to your Teachers Resource Book (TRB) or read on here.

### Communicating and expressing thinking

Being able to explain your mathematical thinking is a very powerful tool, and one that can greatly aid the learning and understanding of both the speaker and the listener(s). Encourage the children to verbalise how they did their calculations (mental or written) to provide you with a window on their thinking. When talking about decimal numbers, encourage children to use fractional language as opposed to decimal language, i.e. ‘6 hundredths plus 4 hundredths is ten hundredths’ etc.

Another way to communicate and express thinking is via jottings. These are informal diagrams that both show and support thinking, and when used as a part-mental approach, serve as an intermediate stage between concrete materials and the abstract calculation. Their use should be encouraged as much as possible (e.g. “use jottings to show me your thinking”) until the child is confident enough to do the whole calculation mentally or using a traditional written form. The main jottings used in Operation Maths are empty number lines (pictured above) and branching (pictured below) to show part–whole relationships and/or explore compensation.

• Dear Family, your Operation Maths Guide to Addition & Subtraction includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• Operation Maths Digital Resources: As always don’t forget to access the linked digital activities on the digital version of the Pupil’s book, available on edcolearning.ie. Tip: look at the footer on the first page of each chapter in the pupil’s book to get a synopsis of what digital resources are available/suggested to use with that particular chapter.
• Mental Maths handbook for Addition and Subtraction from the PDST
• Number Talks book by Sherry Parrish
• Addition & Subtraction Board on Pinterest
• This short video from Graham Fletcher showing the progression of addition and subtraction from the infant classes to the formal written algorithm, with three and four-digit numbers, is also worthwhile viewing:

## Digging Deeper into …. Place Value

Category : Uncategorized

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of place value, please check out the following post: Dear Family, your Operation Maths Guide to Place Value

### Place Value: A Fundamental Concept

When the new school year starts in September, for nearly every pupil from third to sixth class, the first mathematical topic they encounter is place value. This placement is logical; place value is the strand unit from which nearly all of the subsequent number, and measure, strand units build.

On the surface, place value may seem like it is one of the easiest topics to teach; the traditional activities simply involve counting dots on a notation board and/or beads on a place value abacus and, because of this, it is often viewed as an easy topic to kick-start the school year. And, it may appear to the teacher that the children have “got” it…..especially when they are getting all the correct answers in their books. However, it’s usually only later, when difficulties start to arise, often with operations or measures, that the teacher might start wondering “did they really get it?”

Place value is one of THE most important topics in primary mathematics, in that a child’s understanding of the fundamental concepts of place value will greatly impact on their understanding of almost all the other strand units, especially in operations, decimals and measures. Therefore, it is vital that teachers allow sufficient time for the children to explore this topic, moving from experiences with suitable concrete materials (e.g. base ten blocks) to pictorial activities (e.g. drawing base ten materials to represent a given number) and finally to abstract exercises, where the focus is primarily on numbers and/or digits.

That is why Operation Maths has a dedicated block of two weeks devoted to place value in third to sixth classes, and four weeks across the school year in first and second classes, so that there is sufficient time to explore the topic concretely and pictorially. This approach of moving from concrete to pictorial to abstract experiences is generally referred to as a CPA approach.

Indeed, spending sufficient time on meaningful activities now, may reduce potential hurdles later on. Furthermore, revisiting place value activities throughout the year, will allow the children to have ample opportunities to continuously revise and reinforce their understanding. One way to do this is to explore a Number of the Day on a regular basis; use the templates towards the back of the children’s Discovery Books or use the Number of the Day photocopiable in from the Teacher’s Resource Book (TRB). Indeed, a user of Operation Maths reported back to us “I found doing the ‘Number of the day activity’ as often as possible in September is crucial to the ‘Place Value’ chapter”.

### CPA

Concrete materials are key to the children developing a good conceptual understanding of place value. The children need lots of opportunities, in all classes to explore and manipulate a variety of base ten materials. Where suitable/available, these should be introduced in the following order:

• Groupable materials that the children can physically put together in collections of tens and physically take apart. These include lollipop/bundling sticks, straws (counting straws or ordinary drinking straws), unifix/multilink cubes, ten frames and counters etc.

Example of groupable materials: bundling sticks. Example of grouped materials: base ten blocks

• Grouped materials are those already pre-grouped as tens, hundreds, etc., e.g. base ten blocks (also known as Dienes blocks) and/or ten frame flash cards with a pre-set number of dots/counters. These can’t be physically taken apart or combined, instead exchanging/swapping is required.
• Lastly, non-proportional base ten materials.  These are materials that still operate on a base-ten system, but are not proportional to their value i.e. the piece that represents a ten is not ten times the size of the piece representing the unit. Examples include money and place value discs. Money in particular has the advantage that it can also be used to represent decimals i.e. 10c is one tenth and 1c is one hundredth of the unit (euro).

However, money is also limited in that it can only be used to represent numbers up to 999.99. This is where the Operation Maths place value discs become extremely useful. Inspired by similar discs used with Singapore Maths, these cut-outs are included in the free ancillary resources that accompany the scheme and can be used in conjunction with the place value mats on the inside back cover of the Discovery Books. With these discs, it is possible to concretely represent numbers up to 99,999, which had not been possible, using resources available in Ireland, prior to the publication of Operation Maths.

When exploring concrete or pictorial representations of numbers, most children will not have much difficulty interpreting the number once it is presented in the typical, canonical arrangement, i.e. 345 as 3H 4T 5U. However, many children may struggle
to interpret the number correctly if it is presented in a non-canonical arrangement, e.g. 345 as 3H 3T 15U or as 2H 14T 5U. Including activities based on these less common, non-canonical arrangements can encourage children to better understand the relationship between the places and can allow you to better assess the depth of the children’s conceptual understanding, while also preparing them for regrouping.

Operation Maths users can also use the excellent Place Value e-Manipulative, accessible on www.edcolearning.ie. This manipulative can be used to show blocks, straws, money or discs to represent a variety of numbers up to 99,999 and to two decimal places.

2. Click on the Pupil Book icon for your class level.
3. Click on the Edco Resources icon (on book cover image on left-hand side)
4. Select e-Manipulatives from list of categories and then Place Value e-Manipulative.

Read also this post from Beyond Tradition Math showing children representing three-digit numbers in various ways. And watch this video from Origo One on using Numeral Expanders to show expanded form.

### Whole Numbers and Decimal Numbers

Since whole number place value and decimal place value are inherently linked, in Operation Maths for 5th and 6th classes, in the topic of place value the children will explore both whole number and decimal place value together in a very holistic way, thus reinforcing their connectedness, within this strand unit.

While place value understanding includes both whole and decimal numbers, it is important that the children appreciate the differences between them. For example, whole numbers and decimal numbers differ in the variety of correct ways in which they can be written. One ten in standard form is usually written as 10; however, one tenth can be written as 0.1, .1, 0.10, 0.100, etc. For decimals, many teachers often only use one form, usually 0.1, fearing that a variety of ways may confuse children. Conversely, using a variety of ways can actually help reinforce children’s understanding that all of the above forms show one tenth (i.e. a 1 in the tenth place immediately to the right of the decimal point), with most forms (excluding .1) having unnecessary zeros (i.e. in 0.3 the zero is unnecessary; without it the value is still 3 tenths). In other numbers, zero acts as a necessary placeholder between the digits in the neighbouring places; in 30 and .304 the zeros are necessary: without them the values would be 3 units and .34 respectively.

### Verbalising Numbers

For most of our number system, we read numbers in the order that we see the digits, e.g. 345 is three hundred and forty-five. However, the numbers from 11 to 19 are an exception, and as such can present extra difficulties for struggling children. Even for the child who begins to appreciate the meaning of ‘-teen’ as ‘and ten’, the numbers 11 and 12 are additional exceptions to this pattern. Some children may also have difficulties with the -ty numbers (e.g. 120, 130, 140) and in particular may confuse them with the similar -teen number, especially in its verbal form, e.g. fifty vs. fifteen. Even children in the middle and senior classes can struggle to distinguish between the -teen and -ty numbers and it is worth being aware of them as potential hurdles, during this topic.

Regarding decimal numbers, the children can use both decimal language and/or fractional language, i.e. expressing 7.381 as seven point three eight one and also seven and three hundred and eighty one thousandths. Using fractional language to read decimals reinforces the value of the digit(s) in the decimal place(s). However, when using decimal language, it is incorrect to say ‘seven point three hundred and eighty one’, as this refers to hundreds and tens (–ty) which are both whole numbers.

### Number Sense and Visualisation Skills

What is bigger; 12.352 or 12.952? When ordering or comparing, children may use a procedure of comparing digits, which involves examining the digits and realising that both numbers have 1 ten, both have 2 units and the first has only 3 tenths, while the second has 9 tenths, so it is bigger. While this procedure may work successfully, it does not encourage the child to visualise the quantities involved. It would be better for the child to recognise that 12.952 is almost 13 and 12.352 is only a little more than 12 and is therefore smaller. Similarly, with younger classes, it is better for the child to recognise that 52 is just a little more than 50 whereas 58 is almost 60. Asking the children to place the numbers on an empty number line (see below) is an ideal way to promote the development of these visualisation skills. Empty number lines are also a great visual strategy to use when rounding; when rounding 12.352 to the nearest unit we can see that it is between 12.3 and 12.4 so it is closer to 12. The children can use the partial number lines in their Discovery Books as an introduction to this method and then be encouraged to solve the rounding activities in their Pupils’ Books by drawing their own number lines, either on their MWBs or in their copies.

When considering rounding, it it worth noting that it is preferable to use the phrase ‘round(s) to’ as opposed to ‘round(s) up’ and ‘round(s) down’ as these can confuse many children. For example, some people may say the number 69 rounds up to 70, which makes sense since the digit in the tens place has gone up from 6 to 7. However, if a child realises that 43 should be rounded down, they might change it to 30 instead of 40, since that looks more correct to them, i.e. the tens digit has gone down to 3.

For another idea on how to use number lines to aid rounding, please check out this short video from Origo One

Some traditional tasks and activities for place value regularly seen in maths books can give an incorrect picture of a child’s understanding of the concepts of place value. For example, tasks that involve no more than the children identifying the number of identical dots on a notation board, or the number of identical beads on a place value abacus (see opposite), are not good indicators of a child’s understanding of place value, as they are simply demonstrating their number knowledge of numbers to 9. Therefore, these types of tasks have not been included in the Operation Maths series.

### Bigger Numbers

It is vital the children realise that the digits in larger numbers are organised in groups of three with commas as the digit group separators. Insist that the children also use commas in this way, and reiterate that if you can read a three-digit number you can read any size number as long as there are commas present and that you understand the role of the different commas (i.e. one comma means thousands, two shows millions etc.). Interestingly, using commas as digit group separators is a convention largely in English-speaking countries , and other European countries tend to use stops/point or spaces. It is a good idea to highlight this to the older classes, as is done in Operation Maths 6.

It is also worth noting that the concept of the size of a thousand or the size of a million is in itself quite abstract for children. When tackling the large numbers, use all available opportunities to make them real and relevant to children. There is a Maths Around Us video available in the digital resources of Operation Maths 6 made for this purpose; just click on the hyperlink when accessing the digital book.

Also worth noting, is that Operation Maths explores numbers up to 5 digits (whole numbers) in 5th class and through millions in 6th class. However, in the curriculum there is no number limits in 5th and 6th class, therefore the children should not necessarily be limited to these numbers, particularly if they encounter bigger numbers in their environment, books, media, etc. The content in Operation Maths for these classes is deliberately presented in such a way as to encourage children to address numbers above millions where appropriate.

### Place Value in the Environment

As mentioned mentioned previously, it is very important that the children can relate their understanding of place value to numbers around them. To reinforce the relevance of place value, ask the children to collect examples of numbers from the environment. This could include photographs of numbers in the school grounds or locality e.g. car registration numbers, distances on road signs (for Operation Maths 3 users, check out the Maths Trail in the car park, on page 4 of the Discovery Book) . It could also include examples of numbers from print media e.g. newspapers, magazines. In the older classes, you could challenge the children to find an example of a very large number and/or one with the most places of decimals (Operation Maths 6 users, check out the Maths Trail on the internet, on page 6 of the Discovery Book).

What to do with the numbers that the children locate:

• Make a display for the classroom with the examples, in order of size, and giving information for the fact it relates to e.g. the distance to the nearest town etc
• For each example the children find, they must write out the number in word and expanded form (it will likely be in standard form)
• Round the number to the biggest place i.e. if the number is 312 round it to the hundreds, if it’s 0.012 round it to the hundredths etc.

This was the first in a series of “Digging Deeper into …” posts, which will take a more in-depth look at the various topics in primary maths. To ensure you don’t miss out on any future posts, please subscribe to the blog via email, on the top right hand of this page.