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Digging Deeper into … Representing and Interpreting Data (infants to second class)

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

Data Analysis Process

Data analysis, whether at lower primary, upper primary, or even at a more specialised level of statistics, is essentially the same process:

  • It starts with a question, that doesn’t have an obvious and/or immediate answer. Information is then collected relevant to the question.
  • This collected information or data is represented in a structured way that makes it easier to read.
  • This represented data is then examined and compared (interpreted) in such a way as to be able to make statements about what it reveals and, in turn, to possibly answer the initial question (if the question remains unanswered it may be necessary to re-start the process again, perhaps using different methods).

Thus, every data activity should start with a question, for example:

When choosing a question it is worth appreciating that some questions might not lend themselves to rich answers. Take, for example, the first question above; once the data is collected, and represented, there is not that much scope for interpretation of results other than identifying the most common eye/hair colour and comparing the number of children with one colour as being more/less than another colour. However, other questions might lead to richer answers, with more possibilities to collect further information, to make predictions and to create connections with learning in other areas. Take, for example, the question above about travel; the children could be asked to suggest reasons for the results e.g. can they suggest why they think most children walked/came by car on the day in question, whether weather/season/distance from school was a factor and to suggest how the results might be different on another day/time of year. Even in a very simple way, the children are beginning to appreciate that data analysis has a purpose i.e. to collect, represent and interpret information, so as to answer a question.

From Operation Maths Jr Infs TRB p. 147

Sets and Data

Data is very closely related to sorting and classifying sets:

  • The initial question may focus on a particular set in the classroom e.g. identifying the most common/frequent occurring item in the set of farm animals, the set of buttons in our button box, the shoes that the children are wearing, the nature items collected on the walk etc
  • Information is then collected by sorting and classifying the items in the original set according to the target attribute.
  • This collections of items are represented in a structured way that makes it easier to compare e.g. items put in lines of same type, use cubes or drawings to represent the actual items.
  • This represented data is interpreted to answer the question and to make other statements about  relationships e.g. which group has more, less etc

Thus sorting and classify activities should be viewed as potential springboards into data activities and it is important that the children realise that they can represent and compare the size of the sets within each sort by graphing them.

CPA Approach

Even as the children move into first and second classes, it is important that their data activities continue to follow a CPA approach:

Concrete: Continuing to use real objects initially to sort and classify ) e.g. the number of different colour crayons in a box, the different type of PE equipment in the hall , the different fruit we brought for lunch etc), progressing towards using unifix cubes, blocks, cuisinere rods etc to represent the same data. Indeed, the children themselves could be used at this stage; sort the children into groups according to eye colour, hair colour,  age etc and get them to organise themselves into lines that represent the same criterion. This is turn can be very useful for the children to realise that how they are lined up is crucial to being able to interpret the data easily and correctly. If you have visible tiles/markings as flooring on the classroom/hall/corridor, these can be used to organise the “data” accurately!

The children can build block graphs using cubes or blocks, laid flat on a piece of paper or their Operation Maths MWBs.

Pictorial: using multiple copies of identical images to make pictograms and/or using identical cut out squares/rectangles of paper on which the children draw an image that represents the data as it relates to them (e.g. how I traveled to school today). These can then be collected and organised into lines, so that it is easier to read the data. As a development, identical cut out squares/rectangles of paper of different colours can be used with the children taking the correct colour as it relates to them (e.g. choosing the colour for their eyes/hair colour etc.) while also progressing towards using a specific colour for a specific criterion (“Take a blue square if you walked to school today”). Thus, the children should begin to appreciate the need to label the graph, axes etc so that the meaning of the represented data can be correctly interpreted.

HINT: A common confusion among children when making vertical graphs of any type is that the pictures/blocks start at the top and go down; an understandable misconception when you consider that in most other activities we work from the top down! A simple way to show how vertical graphs are formed, is to demonstrate, using a concrete Connect 4 type game, how the first counter in each column falls to the bottom and subsequent counters in that column build up from there. If you don’t have an actual Connect 4 game in your classroom you could use an interactive type such as this one here. 

Abstract: the final stage, where the focus is primarily on numbers and/or digits e.g. identifying how many, how many more prefer this than that etc.

Further Reading and Resources

Thinking Strategies for Multiplication and Division Number Facts

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What are number facts?

Number facts are the basic number facts that, it is hoped, children could recall instantly, so as to improve their ability to compute mentally and use written algorithms. Traditionally referred to as tables, the multiplication and division number facts typically include all the multiplication facts up to 10 x 10 and their inverse division sentences.

Some of the big ideas about number facts:

  • Some facts are easier than others to recall – which ones, do you think?
  • The easier facts can be used to calculate other facts – which ones, do you think?
  • The same fact can be calculated using various approaches – these approaches are often referred to as thinking strategies – see more below.
  • Using thinking strategies means that the children can apply the understanding, to facts beyond the traditional limits of “tables”.

 

What are thinking strategies?

A thinking strategy is a way to think about a process to arrive efficiently at an answer. For example, if asked to multiply a number by 2, one could double the number. Doubling is a very effective thinking strategy for the multiplication facts of 2, 4 and 8, as can be seen in the video below.

 

Halving is the opposite to doubling. And halving is a very effective thinking strategy to use for the multiplication facts of 5; if asked to multiply a number by 5, one could think of 10 times the number and then halve that amount (see below).

The Operation Maths  and Number Facts books for third and fourth classes repeatedly emphasise (among other thinking strategies) the strategy of doubling and halving known facts to derive unknown facts, eg through doubling I can work out 2 times, 4 times and 8 times a number; if I know 10 times the number I can work out 5 times, etc. 

From Operation Maths 3, possible thinking strategies for 2x, 5x, 10x.

The 100 dots grids on the inside back covers of Operation Maths 3 and 4 and Number Facts 3 and 4 can be extremely useful for the pupils to model various arrangements/arrays, while the teacher can use the Operation Maths 100 square eManipulative to replicate (and label) the children’s arrangements on the IWB.

Using doubling to model 2 x 6, “2 rows of 6”, 4 x 6, 8 x 6 (left) and trebling to model 3 x 7, 6 x 7, 9 x 7 (right)

Furthermore, multiplication and division are taught together throughout the Operation Maths series, so that, rather than compartmentalising each operation, the children develop a better understanding of how both concepts relate to each other. In this way, the basic division facts are easier to acquire, as they are understood to be the inverse of the more familiar multiplication facts. However, it is important that within each group of facts, the children explore the multiplication facts first; the better their understanding of these, the more likely they are understand the inverse division facts. Indeed, “think multiplication” is in itself, a thinking strategy for the division facts (see video below).

 

Traditionally, learning “tables” had been by rote, but current research suggests that this is ineffective for the majority of children. In contrast, children should be taught to visualise numbers and to use concrete materials, images and thinking strategies to use what they know to solve what they do not know. Below are examples of some useful thinking strategies for the basic multiplication and division facts (taken from Number Facts 3 & 4, Edco, 2018)

There can often be different ways to think about the same fact (or groups of facts), and the children should always be encouraged both to identify alternative approaches and to choose their preferred strategy. For example, consider 5 x 9:

5 times is half of 10 times: 10 × 9 = 90, so 5 × 9 = half of 90 = 45
9 times is one set less than 10 times: 10 × 5 = 50, so 9 × 5 = 50 − 5 = 45
9 times is treble 3 times: 3 × 5 = 15, so 9 × 5 = treble 15 = 45

Once the children understand how to arrive at an answer via a thinking strategy, they can then apply this thinking strategy to more complex calculations that are beyond the traditional 10 x 10 ceiling of “tables”; for example if I understand 5 times any number is half 10 times the number, then I can use this to mentally calculate 5 x 18, 5 x 26 etc (see more on this below).

 

Computational Fluency:

‘Fluency requires the children to be accurate, efficient and flexible.’ (Russell, 2000).

The primary aim of both the Operation Maths and Number Facts series (see more information on Number Facts below) is to enable the children to become computationally fluent. To achieve computational fluency, the children must be accurate, efficient and flexible:

  • Accurate: the children must arrive at the correct answer, e.g. 6 x 8  =48.
  • Efficient: the children must calculate the answer in an efficiently. A child who produces an answer of 48 in response to the question 6 × 8 by counting in jumps of six or eight may be accurate but is not efficient.
  • Flexible: children must be able to visualise and mentally manipulate numbers in order to see how they might be broken down and recombined to get an accurate and efficient answer (as shown with the various ways to consider 6 x 8 below).

Thus, flexibility is the key to fluency. A child who only knows that 6 x 8 = 48 becasue they have memorized that fact, is missing out on all the various possible connections between those numbers, subsequently hampering future connection-building. In contrast, a child who is flexible with number facts is one with a well-developed number sense, who can see the connections both between and within numbers, i.e. they can partition and/or combine numbers into more compatible (friendly) amounts and can apply their strategies to numbers beyond those they have dealt with. Therefore, a thinking strategies approach will not only be effective for aiding understanding and recall of the basic facts up to 10 x 10, a thinking strategies approach can enable children to apply these mental computation skills to numbers beyond this traditional ceiling, as shown below.

From Number Facts 4

 

The Number Facts Series from Edco

Number Facts is latest addition to the Edco Primary Maths stable, and it is a series of activity books designed to foster fluency in number facts for primary school children from First Class. The series features an innovative approach to the acquisition of basic number facts, and, like Operation Maths, teaches children to understand, not just do, maths.

Image result for number facts edco

In contrast to the more traditional drill-and-practice workbooks, which just test whether the answer is known, Number Facts teaches children to visualise numbers pictorially and to use these images and thinking strategies to become more adept at manipulating numbers. The specific focus of Number Facts will be to develop children’s thinking strategies and apply these to the basic number facts in such a way as to promote the child’s ability to visualise and recall these facts, thereby achieving fluency.

Both this rationale, and the suggested teaching approaches to the teaching of the basic multiplication and division facts for third and fourth classes, are clearly outlined in the Teachers Resource Book (TRB) which accompanies the series, and which is downloadable here. This TRB also includes a Long Term Plan for both third and fourth classes (see extract below), outlining a logical progression for the various fact groups throughout the school year. To view sample pages from the pupils Number Facts books please click here. Sample copies of all the books are also available from your local Edco reps.

 

Further reading and viewing:

 

 

Maths by Month – October (updated 2018)

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Welcome to the second installment in this year’s series of posts designed to explore the Operation Maths topics on a month-by-month basis, giving teachers greater insights into the concepts at hand, when they are most relevant. While each monthly overview will specifically zone in on the Operation Maths topics for that particular month, the information and suggestions will be relevant to ALL primary teachers, whether they are Operation Maths users or not.

HINT: To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the Operation Maths blog via email, on the top right hand of this page.
Another way to keep up to date an all new maths-related developments is to like/follow the Edco Primary Maths page on Facebook and/or Twitter 

Operation Maths for Junior Infants to Sixth Class:

  • Operation Maths users can also access a class specific, month-by-month list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 
    1. Log into your edcolearning account
    2. Click on the At School Book/Pupil’s Book for your class level.
    3. Click on the Edco Resources icon (on book cover image on left-hand side)
    4. Select Weblinks from list of categories and then click to download the document.
  • Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above.

HINT: If you are new to Operation Maths this year or have changed class level, be sure to check out the Quick Start Guide to the Operation Maths books and the companion Quick Start Guide to the Operation Maths Digital Resources
Don’t forget that Operation Maths also has you covered for planning whether you’re teaching a single class or multi-class. 

Other suggestions for October:

  • The October plan for third to sixth classes has deliberately allowed for a free week, to enable teachers to engage with Maths Week, held every year at this time. This year, Maths Week will run from 13-21 October. Make sure to register your school at the link above and then organise some fun maths activities for your class or school. You can follow the links in the site to find out more about Maths mazes, Maths Art (which, coincidentally, links very well with October Operation Maths for 3rd and 4th classes i.e. tessellations in 2D shapes), Maths and history,  code breaking and lots more.
  • You could also make Maths Week become a game-themed week in your class. Teachers of third to sixth classes could use the Games Bank in the Operation Maths TRB. Teachers of infants to second classes can use any of the games listed in the short-term plans in their TRBs.
  • Another option for Maths Week, if you didn’t already do it in September,  is Jo Boaler’s  Week of Inspirational Maths. Click on the link for an overview of the activities in Week of Inspirational Math, and scroll down to the bottom of the page to access all the resources; K-2 roughly align with Infants to 2nd and Grades 3-5 roughly align with 3rd-6th classes.

We’re here to help!
If you have any questions on Operation Maths, Number Facts or anything related to primary maths over the course of the school year, please PM or contact Edco Primary Maths via Facebook and/or Twitter 

Digging Deeper into … Addition and Subtraction (infants to second class)

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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

A quick look at the maths curriculum for junior and senior infants will reveal that, within the strand of number, there are no strand units entitled operations, addition or subtraction, as are evident in the curriculum for first and second classes. However both operations are there – under the guise of combining, partitioning and comparing.

Addition and subtraction are two of the four basic mathematical operations (multiplication and division being the other two):

  • Addition involves the joining/combining of two or more quantities/sets/parts to get one quantity/whole/set, typically referred to as the sum or total. There are two main types: active (2 children at a table and 3 more join them) or static (2 boys and 3 girls at a table, how many children in all?)
  • There are three types of subtraction:
    • take-away (active) which involves the removal/deduction of one quantity/part from a whole amount/quantity
    • comparison (static) which involves identifying by how much one quantity/set is more or less than another (the difference)
    • missing addend (active) which involves identifying the amount needed to combine with a known part to make a whole.

In each type of subtraction we know the total/whole and a part and we need to find the missing part, which could be the amount left, the difference or the missing addend.

The types of addition and subtraction are explained very clearly and succinctly in the Origo One videos below.

 

 

Relationship between addition and subtraction

As shown in the videos above, addition and subtraction are inverse operations; we can demonstrate addition by adding more to an existing amount; the reverse action would involve removing an amount, thus demonstrating subtraction as take away. In contrast to traditional maths schemes, which often have separate chapters for each of these operations, Operation Maths predominantly 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.

Beginning in first and second classes, the children are enabled to understand addition and subtraction as being the inverse of each other, which will progress towards using the inverse operation to check calculations in higher classes.

 

CPA Approach within a context

As mentioned repeatedly in previous posts, both the Operation Maths and Number Facts series are based on a CPA approach. Furthermore, as was referenced in the videos above, for the children to develop a deep understanding of the different types of addition and subtraction, there has to be some context or story, with which they can identify. This, in turn, should be explored via progression through concrete, pictorial and abstract stages.

This context can be simply made up by the teacher or be inspired by a picture book that the class is reading. It can be modeled using the concrete materials available in the classroom (eg plastic animals, toy cars, play food etc. ) and/or using the Operation Maths Sorting eManipulative (see below) and the extensive suite of inbuilt images; the images can be shown either with or without a background (background options include five and ten frames, set outlines and various grids).

HINT: To find out more about how to use the 5, 10 and 20 frames that accompany the Operation Maths series please read on here: http://operationmaths.ie/youve-been-framed-closer-look-ten-frames/

As the children progress, the need arises to record the operations using some graphic means. Initially, this can include representing each of the items in the story with counters and/or cubes. In turn, bar models could also be used to represent number relationships, while bearing in mind that different types of bar models will be required to model different context and types of subtraction (even though the subtraction sentences, if using them, might look exactly the same). Using the examples below, the first bar model (a part-whole bar model) could be used to model this story: Snow White had seven dwarfs. If four of these went to work, how many were left at home? Whereas, the second bar model (a comparison bar model) would better suit this story: the seven dwarfs all wanted to sit down at the table but there were only four chairs. How many dwarfs had no chair?

While bar models do not specifically appear until in the pupils books until Operation Maths 3, the children could use and explore simple bar models. Thinking Blocks Jr is based on simple bar models and could be shown to the class on the IWB while the children suggest answers and labels on their Operation Maths MWBs.  Then the children could draw simple models in their books to help solve the word problems there. Furthermore , as shown above, the Bar Modelling eManipulative could also be used to create bars of different length.

Before rushing too quickly into abstract recording (using only digits and symbols), an alternative intermediary stage could be to represent the relationships, using a branching bond (opposite). Similar to the part-whole bar model earlier, this branching bond structure encourages the children to appreciate that two sets/parts ( 4 and 6) can be combined to make a larger set/whole (10). Inversely, when a part (4) is removed from the whole (10), a part is left (6). This bond structure can also represent the missing addend type of subtraction: if a part was hidden (6), the question could be asked  what must be added to 4 to make 10.

Both branching bonds and simple bar models are used throughout the Number Facts series to represent relationships and demonstrate strategies. They are also used throughout the Operation Maths 3-6 books, but in increasingly more complex situations.

 

The meaning of the equals sign

With the formal introduction of addition number sentences in senior infants (ie the recording of relationships using the plus and equals sign), followed by the formal introduction of subtraction sentences (using the minus sign) in first class, comes the need to correctly interpret the purpose of the equals sign as identifying equivalence; ie that the value on one side of the equals sign is the same as the value on the other side. It is essential at this stage that the children don’t interpret the equals signal incorrectly as being a signpost indicating that the answer is coming next. A pan or bucket balance is an extremely valuable resource to help demonstrate equivalency, as can be seen in the video below.

Calculations in the Operation Maths book are often shown vertically and horizontally. When presented horizontally, it is often misinterpreted that the children must now rewrite the calculation vertically, to be solved using the traditional column method (see more on the column method in the next section). Rather, presenting calculations horizontally is a deliberate effort to encourage the children to explore how to solve the calculation using a concrete based approach and/or using a mental strategy, as opposed to always tackling these calculations in a written way.

 

Looking at more complex numbers

In first and second classes, once introduced to operations using two-digit numbers, children can often have tunnel vision (or column vision) regarding addition and subtraction calculations: they “do” the units, and then the tens, 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 CPA approach. This means that the children’s initial experiences should involve groupable base ten concrete materials (e.g. bundling straws or lollipop sticks, ten-frames and counters, unifix or multi-link cubes arranges in sticks of ten, see below), where a ten can be physically decomposed  into ten units and vice versa, before moving on to pregrouped base ten materials (eg base ten blocks/Dienes blocks, base ten money and/or Operation Maths place value discs) which require a swap to exchange a ten for ten units and vice versa.

When children are comfortable with the manipulating the concrete materials, they can move on to a process whereby these materials are represented pictorially and/or demonstrate the process using a suitable the visual structure eg an empty number line and/or bar model. Abstract exercises, where the focus is primarily on numbers and/or digits, should only appear as part of the final stage of this process.

When exchanging tens and units (or tens and hundreds in second class), reinforce that a ten is also the same as 10 units, and that a hundred is the same as 10 tens and is the same as 100 units. The use of non-canonical arrangements of numbers (e.g. representing 145 as 1H 3T 15U or  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.

Mental strategies are as important as written methods

In first and second classes, 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, without really looking at the entire numbers or the processes involved. Therefore, while the column method for addition and subtraction is an important aspect of this topic, equally important is the development of mental calculation skills, via a thinking strategies approach.

From Number Facts 1 & 2

Thus, one of the main purposes of the operation chapters in Operation Maths is to extend the range of strategies that 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.

To find out more about using a thinking strategies approach to teach the basic addition and subtraction facts please read on here.

When meeting new calculations, ask the children, as often as possible, 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.

HINT: Number Talks are a fabulous resource to use alongside the Operation Maths and/or Number Facts series, as they complement their thinking strategies approach. Read on here to find out more about where both Operation Maths and Number Talks overlap.

 

Key messages:

  • There are different types of addition and subtraction and children need to explore the different types to gain a deep understanding of the concepts
  • As children encounter new numbers and new number ranges, be it numbers to ten in infants, teen numbers to 199 in first and second classes, they should be afforded ample opportunities to combine to make these amounts, partition these amounts and compare these amounts using concrete materials and via some story-like context.
  • Initial recording of these relationships should be via counters and cubes etc, before moving on to pictorial representations of the same and/or using frames, maths rack, bar models, branching bonds etc.
  • Addition and subtraction number sentences, that use only digits and symbols, should be avoided until the children demonstrate readiness for this more abstract stage.
  • Encourage the children to use and develop mental strategies and avoid focussing almost exclusively on the formal, traditional ways of doing addition and subtraction ie column method.

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 very worthwhile viewing and summarises the key messages well.

Further Reading and Resources:

  • Dear Family, your Operation Maths Guide to Addition and 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.
  • For more hints and tips specific to each class level, check out the “What to look out for” section in the introduction to this topic in the Teacher’s Resource Book (TRB)
  • Number Talks book by Sherry Parrish
  • Mental Maths handbook for Addition and Subtraction from the PDST
  • Splat! Similar to Number Talks, these free resources from Steve Wyborney encourage discussion and reasoning. Play the PowerPoint presentations on your class IWB while the children use their Operation Maths MWBs to respond.
  • Addition & Subtraction Board on Pinterest

Thinking Strategies for Addition and Subtraction Number Facts

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What are number facts?

Number facts are the basic number facts that, it is hoped, children could recall instantly, so as to improve their ability to compute mentally and use written algorithms. Traditionally referred to as tables, the addition and subtraction number facts typically include all the addition facts up to 10 + 10 and their inverse subtraction sentences.

Some of the big ideas about number facts:

  • Some facts are easier than others to recall – which ones, do you think?
  • The easier facts can be used to calculate other facts – which ones, do you think?
  • The same fact can be calculated using various approaches – these approaches are often referred to as thinking strategies – see more below.
  • Using thinking strategies means that the children can apply the understanding, to facts beyond the traditional limits of “tables”.

What are thinking strategies?

A thinking strategy is a way to think about a process to arrive efficiently at an answer. For example, if asked to add 9 to a number, one could think of moving 1 from the other addend to the 9 so as to make a 10, which therefore becomes an easier calculation (see below)

      

The Operation Maths books for first and second classes emphasise three specific thinking strategies throughout: counting on from the biggest number, using doubles and near doubles and using the number bonds for ten (see image below). The doubles facts and bonds of ten are also included on the pull-out flap at the back cover to the pupils books, both for quick reference and to emphasise their importance.

From Operation Maths 2 At School Book

In the case of doubles, near doubles and bonds of ten, these key sets of number facts tend to be easier for children to understand and recall. These facts also make up a core section of the total addition facts to 10 + 10, as highlighted below on the addition square. When these become known facts, they can then in turn be used to calculate unknown facts (eg if 7 + 3  = 10, then 7 + 4 = 11), thus covering an even greater number of the total addition facts.

Furthermore, addition and subtraction are taught together throughout the Operation Maths series, so that, rather than compartmentalising each operation, the children develop a better understanding of how both concepts relate to each other. In this way, the basic subtraction facts are easier to acquire, as they are understood to be the inverse of the more familiar addition facts.

Traditionally, learning “tables” had been by rote, but current research suggests that this is ineffective for the majority of children. In contrast, children should be taught to visualise numbers and to use concrete materials, images and thinking strategies to use what they know to solve what they do not know. Below are examples of some useful thinking strategies for the basic addition and subtraction facts (taken from Number Facts 1 & 2, Edco, 2018)

From Number Facts 1 & 2

There can often be different ways to think about the same fact (or groups of facts), and the children should always be encouraged both to identify alternative approaches and to choose their preferred strategy. For example:

8 + 6 = (5 + 3) + (5 + 1) = 10 + 4 (make a ten) = 14
8 + 6 = 10 + 4 (move 2 from 6 to 8 to make a ten) = 14
8 + 6 = 7 + 7 (move 1 from 8 to 6 to make a double) = 14

Once the children understand how to arrive at an answer via a thinking strategy, they can then apply this thinking strategy to more complex calculations that are beyond the traditional 10 + 10 ceiling of “tables”; for example if I understand different ways to calculate that 8 + 6 = 14, then I can use these ways to mentally calculate 18 + 6 , 18 + 16 etc.

Computational Fluency:

‘Fluency requires the children to be accurate, efficient and flexible.’ (Russell, 2000).

The primary aim of both the Operation Maths and Number Facts series (see more information on Number Facts below) is to enable the children to become computationally fluent. To achieve computational fluency, the children must be accurate, efficient and flexible:

  • Accurate: the children must arrive at the correct answer, e.g. 8 + 6 = 14.
  • Efficient: the children must calculate the answer in an efficiently. A child who produces an answer of 14 in response to the question 8 + 6 by ‘counting all’ (eg have to count up to a total using using counters, fingers, etc.) may be accurate but is not efficient.
  • Flexible: children must be able to visualise and mentally manipulate numbers in order to see how they might be broken down and recombined to get an accurate and efficient answer (as shown with the various ways to consider 8 + 6 above).

Thus, flexibility is the key to fluency. A child who only knows that 8 + 6 = 14 becasue they have memorized that fact, is missing out on all the various possible connections between those numbers, subsequently hampering future connection-building. In contrast, a child who is flexible with number facts is one with a well-developed number sense, who can see the connections both between and within numbers, i.e. they can partition and/or combine numbers into more compatible (friendly) amounts and can apply their strategies to numbers beyond those they have dealt with. Thus, a thinking strategies approach will not only be effective for aiding understanding and recall of the basic facts up to 10 + 10, a thinking strategies approach can enable children to apply these mental computation skills to numbers beyond this traditional ceiling e.g. 19+ 5, 29 + 6 etc (see below).

       

The Number Facts Series from Edco

Number Facts is latest addition to the Edco Primary Maths stable, and it is a series of activity books designed to foster fluency in number facts for primary school children from First Class. The series features an innovative approach to the acquisition of basic number facts, and, like Operation Maths, teaches children to understand, not just do, maths.

Image result for number facts edco

In contrast to the more traditional drill-and-practice workbooks, which just test whether the answer is known, Number Facts teaches children to visualise numbers pictorially and to use these images and thinking strategies to become more adept at manipulating numbers. The specific focus of Number Facts will be to develop children’s thinking strategies and apply these to the basic number facts in such a way as to promote the child’s ability to visualise and recall these facts, thereby achieving fluency.

Both this rationale, and the suggested teaching approaches to the teaching of the basic addition and subtractions facts for first and second classes, are clearly outlined in the Teachers Resource Book (TRB) which accompanies the series, and which is downloadable here. This TRB also includes a Long Term Plan for both first and second classes (see extract below), outlining a logical progression for the various fact groups throughout the school year. To view sample pages from the pupils Number Facts books please click here. Sample copies of all the books are also available from your local Edco reps.

Further reading and viewing:

  • Are you compensating? A closer look at the thinking strategy of compensation.
  • Number Talks : this is a maths methodology centered around the development of  strategies and mental calculation skills. As such, it really complements both the Operation Maths and Number Facts series. For more information on where Operation Maths and Number Talks overlap, please read on here.
  • Mental Maths handbook for Addition and Subtraction from the PDST
  • Number Facts Board on Pinterest
  • The Origo One videos below are a great way to get an overview of some various thinking strategies, each in 60 seconds or less!

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

Maths by Month – September (updated 2018)

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It’s a new school year! And with it comes the first in this year’s series of posts designed to support teachers on a month-by-month basis, giving teachers greater insights into the concepts at hand, when they are most relevant. While each monthly overview will specifically zone in on the Operation Maths topics for that particular month, the information and suggestions will be relevant to ALL primary teachers, whether they are Operation Maths users or not.

HINT: To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the Operation Maths blog via email, on the top right hand of this page.
Another way to keep up to date an all new maths-related developments is to like/follow the Edco Primary Maths page on Facebook and/or Twitter 

Operation Maths for Junior Infants to Sixth Class:

  • Operation Maths users can also access a class-specific list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 
    1. Log into your edcolearning account
    2. Click on the At School Book for your class level.
    3. Click on the Edco Resources icon (on book cover image on left-hand side)
    4. Select Weblinks from list of categories and then click to download the document.
  • Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above.

HINT: If you are new to Operation Maths this year or have changed class level, be sure to check out the Quick Start Guide to the Operation Maths books and the companion Quick Start Guide to the Operation Maths Digital Resources
Don’t forget that Operation Maths also has you covered for planning whether you’re teaching a single class or multi-class. 

Other suggestions for September:

  • Check out the “Maths and me” attitudes questionnaire, situated after the last assessment in the Operation Maths Pupil Assessment booklet for 3rd to 6th classes. Suggest to the children that they fill this using a particular colour on one of the first days of the school year to be then revisited later in the year. At this point, the children can again record their attitudes in a different colour and reflect upon any changes they made, if any.
  • Maths about me: another great activity for the start of a new school year. The children write facts about themselves that are appropriate to their ability eg height, age, shoe size, telephone number, distance from school (use google maps), time that they get up or go to bed etc. This can be recorded on the inside front cover of the discovery book, filled in on a pre-made template from the internet, used to make a large class display or even become a more complex problem solving activity in the more senior classes.
  • Inspire your class for the year ahead: Most people have this belief that there is such a thing as a maths brain, a belief which Jo Boaler, among others, strongly challenges. In conjunction with her youcubed team at Stanford University, in 2015 they put together resources, videos etc for a Week of Inspirational Maths and followed that up with a Week of Inspirational Maths 2 and 3 in 2016 and 2017 respectively. The latter two have lessons and activities aimed at infants to 6th, as well as second level. Click on the link for an overview of the activities in Week of Inspirational Math, and scroll down to the bottom of the page to access all the resources; K-2 roughly align with Infants to 2nd and Grades 3-5 roughly align with 3rd-6th classes.
  • New year, new initiative! Number Talks is an excellent maths methodology that is gaining traction globally, and more recently, nationally thanks to the promotion of the PDST. Better still, the rationale behind it aligns itself very closely with the underlying principle of Operation Maths, that is teaching children to understand maths, not just do maths. To find out more about number talks and to access a whole suite of ready-made resources for all class levels just click on the link above.

We’re here to help!
If you have any questions on Operation Maths, Number Facts or anything related to primary maths over the course of the school year, please PM or contact Edco Primary Maths via Facebook and/or Twitter 

Digging Deeper into … Counting and Numeration

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of counting and numeration, please check out the following post: Dear Family, your Operation Maths Guide to Counting and Numeration.

Counting and numeration are listed as strand units in the strand of number for Junior Infants, Senior Infants, First and Second Class in the Primary Maths Curriculum (1999) and counting and numeration at each of these classes require similar skills, although the range of numbers will differ. However, while counting and numeration is specified as strand units only in infants to second, the understanding required is just as relevant and as important in the higher classes e.g. counting with larger numbers, counting fractions, decimals, percentages, etc.

Learning to count: rote versus rational counting

You are probably all familiar with the scenario: a parent declares that their pre-school age child can count because they can rattle off numbers to ten! As we all know, counting involves much more that just listing off numbers (rote counting). Watch this one minute video, which synopsises the difference between rote and rational counting.

While rote counting is relevant when learning to count, to count with understanding (i.e. rational counting) depends on the child developing an appreciation of rational counting, via the five counting principles, (briefly outlined in the video above); each of these counting principles are explained further in these follow-on videos from Origo Education:

HINT: For more information on the Counting Principles, including suggestions on what to look out for and what to ask/do, check out this blog post.

Apart from rattling off numbers, a child’s main interest in counting is to identify the quantity of objects in a set. “How many cars do you have? I have six cars”. Cardinality is using counting to find out “how many”.  And, since most of the sets that children will encounter, and will want to count, will be randomly arranged, then teaching the order-irrelevance principle will probably be most relevant to the children themselves. Therefore, the children must develop some strategies to ensure that they count every object, once only:

  • Count and tag: as each item is counted it is touched (this works quite well if the set to be counted is already in a line, or a rectangular array, but doesn’t work as well with scattered sets .
  • Count and push/put: as each item is counted it is pushed to the side or put into a pot, tray etc.
  • Count and mark: put a mark beside each item as it is counted; this works well for pictorial representations that cannot be physically moved.
  • Count and group: in the case of large collections (for example in first and second classes), rearrange the objects into “friendly” groups (eg two, tens or fives) that the children can easily skip-count. Using the Operation Maths frames and structures to help to reorganise the objects can be of particular benefit.

This ability to demonstrate one-to-one counting should not be taken for granted; while it seems quite a simple concept, many children can struggle. Therefore, when the focus is on the cardinality of counting (establishing how many), all counting activities should be counting something; lining toys up and counting how many by tagging each one, etc.

When observing children as they count, check:

  • Do they “tag” each object as they count (eg pushing them aside)?
  • Can they count regular arrays or rows?
  • Can they count random groups in some sort of systematic way so that they don’t miss or double up on objects?
  • Can they count the same set several times, starting with a different object each time?
  • Can they show how rearranging the objects does not change the quantity?

HINT: use relevant number rhymes and stories to reinforce counting and number word sequence. Many of the short-term plans (STPs) in the Operation Maths TRBs list various possibilities; see the Literacy suggestions in section on Integration

Counting without Counting!

When can you count without counting? When you subitise! Subitising is the ability to recognise a quantity at a glance, without counting. When you throw a five on a die, usually it is not necessary to count the individuals dots; we recognise that there are five dots from their shape. So, while it is very important that we spend significant time practising one-to one counting initially, this is not the most efficient approach, and we do want the children to progress to a point where they do not need to count each item/object individually.

Ways to promote subitising:

  • Play lots of dice and domino games; the Operation Maths TRBs have game suggestions and station activities in every STP plan, many of which are based around dice etc.
  • Use the Operation Maths frames: the visual layout of various numbers in the frames (see image below) encourages the children to internalise a picture of how the numbers look and to recognise this in other situations.
  • Play dot flash: briefly show the children dot cards in various arrangements and ask them to tell you what they saw. There are photocopiable dot cards at the back of the Operation Maths TRBs for this purpose.
  • Use other structures that have a definite layout eg rekenreks (or maths rack) can also be used. This visual structure features quite strongly in the Number Talks presentations for junior infants, senior infants and first class, all available at the link above.
  • Arrangements of Base Ten blocks, bundled sticks and/or place value discs can also be used.
  • Use online games (eg Number Flash from Fuel the Brain) and/or suitable apps (such as this free one)

HINT: For more suggested subitising activities read this blog post Counting With Your Eyes

Numeration

Numeration involves the children being able to match a numeral and its matching number word to each other and to various different arrangements of objects (both identical and non-identical) of that amount eg 3 = three = 🏀⚾⚽ = 🚗🚗🚗.

As the children move into first and second classes, numeration will move beyond the numbers to ten, through the teen numbers and all the way up to 199. Numeration in these classes involves much more than just matching a quantity to the numeral and to the number word:

  • The children need to appreciate the visual pattern of numbers in sequence: 20, 21, 22, 23, 24, 25, 26…
  • The children need to recognise the patterns in the number word sequence when spoken: “twenty one, twenty two, twenty three, twenty four…”
  • From this understanding the children should be able to count forwards and backwards from various starting points. They should also be able to identify the number before or after a given number.

Visual structures, such as the Operation Maths 100 Square e-Manipulative (see below), can be very useful, as:

  • they provide the numbers in order
  • the patterns can be easily identified
  • individual squares and/or large sections can be hidden and then revealed for the children to test their ability to identify preceding and subsequent numbers in a sequence.

HINT: Particular attention should be given to the multiples of ten ie the “ty” numbers and a deliberate distinction should be made between the “ty” numbers and the “teen” numbers, especially when being verbalised i.e. there is little difference verbally between eighteen and eighty, but there is a significant difference between these numbers in value . Like the “teen” numbers, “ty” numbers are also widely acknowledged as common hurdles for children and so time spent now will be time well spent for the future. 

Further Reading and Resources:

This is part of the series “Digging Deeper into …” which takes 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.

Digging Deeper into … Early Mathematical Activities

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of Early Mathematical Activities, please check out the following post: Dear Family, your Operation Maths Guide to Early Mathematical Activities. 

Early Mathematical Activities (EMA) is a strand in the Primary Mathematics Curriculum (1999) for children in junior infants only, although the activities might also be suitable for children in senior infants as revision, as well as being suitable for many children in their final preschool year.

It includes the strand units of:

  • Classifying
  • Matching
  • Comparing
  • Ordering

While comparing and ordering appears as a strand unit also in the strand of Number, for EMA the emphasis should not be on using number or counting to describe relationships, rather on the attributes themselves. However, once the children have been introduced to the numbers,  the early mathematical activities can be repeated, but now to include using the opportunities presented to incorporate numbers or counting to describe relationships.

Sets

EMA is fundamentally all about sets; a set is any collection that has been grouped together in some meaningful way. Sets are all around us, and much of a young child’s exploration of the world involves the child seeing things in terms of sets e.g. my toys, the set of toys that belong to me as opposed to all other toys. Sets are also fundamental to developing an understanding of number and operations: numbers are used to describe the quantity in a set;  a quantity will be removed from a set to model subtraction etc.

Matching

Although classifying is listed before matching in the Primary Mathematics Curriculum (1999), matching is actually less complex, as typically we understand matching as completing a pair, whereas classifying is typically interpreted as organising a collection into two or more subsets. For this reason, in Operation Maths for junior infants, the children first match pairs of identical objects (reinforcing one-to-one correspondence), using the language same for those that match and different for those that don’t match.

  • Start with a limited amount of objects e.g. eight, where each has a match that is the same (ie fully alike). This can be demonstrated using real objects and/or on the class IWB, using a representation of real objects, using the Operation Maths Sorting e-Manipulative (see image above).
  • Working with a small group of children, isolate one of the objects and ask a child to “find the match/find the one that is the same” i.e. identify the other that is fully alike, and most importantly, to verbalise why it is the same and therefore the correct match. In this way, you are asking them to justify their choice using the language of the attributes.
  • The children can also be asked to orally justify why certain objects are not the same/are different.
  • Initially, you can use objects where each pair is completely unlike the other pairs e.g. four different shapes, in four different colours. Then, progress towards collections where, while there are like objects, there is only one match that is fully alike/exactly the same (see example above).
  • To make this task more complex:
    • introduce more attributes (e.g. size) and a larger range of attributes (e.g. more shapes and colours).
    • increase the size of the collection
    • remove/conceal an object and ask the children to identify the object which now has no match and to use this to be able to describe the object that has been removed/concealed.
    • See also the Clothesline activity in the Junior Infants TRB (p 16). The materials could be expanded to include gloves as well as socks. Initially try to ensure that there is two of everything (in order to make complete pairs). When the children locate the matching items, they should explain why they are the same, before hanging them up using clothes pegs (clothes peg activities have the added advantage of developing pincer grip and fine motor skills necessary for correct pencil grip). A development of this activity if the teacher deems it suitable: include odd socks/gloves and observe how children react. Use questioning to elicit their own “rules” for dealing with these and how they might describe them. If appropriate, use the opportunity to discuss and introduce language such as even, odd etc., if the children do not suggest this terminology themselves.
    • For further experience using one-to-one correspondence, the children should also have opportunities to match pairs of related objects,  i.e. objects that are not the same, but that purposely go together,  e.g. putting out knives and forks, buttoning coats, putting lids on boxes/tubs. Again, many of these activities, using objects from the children’s daily lives, will also be useful for developing and strengthening fine motor skills.

HINT: Commercial products such as attribute bears, shapes and people are very useful for all EMA and may appear to even be the most suitable material because the attributes, and therefore the “rules” that govern a set can be deciphered clearly. However, they can also be limited, in that there is little negotiation required. Thus more arbitrary materials, such as items from nature (stones, rocks, leaves etc), children’s own clothing items (socks, shoes, gloves) and assorted toys (threading beads, toy cars, soft toys etc) and indeed any objects in the classroom for which there is at least one other that is fully alike, can provide greater opportunities for mathematical discussion and thinking, as the children have to come up with their own ways to group them. In particular, see the Aistear play suggestions in the Operation Maths TRB for Junior Infants.

Classifying

Classifying (or sorting) is different from matching as classifying involves reorganising a collection into two or more subsets. When presented with a large set of objects e.g. toys, children will often isolate a certain group of objects e.g. take out all the toy cars. In this way they have made a set of cars and (by default) a set that is not cars. This is referred to as a binary sort, where two subsets have been created: one which has the chosen attribute and one which does not. In mathematical terms this “opposite” set is the complement of the chosen set.

  • The children should have lots of opportunities to explore various collections of objects from which they will likely create their own sets. Through questioning, elicit from the children an explanation (ie rule) for their set.
  • The teacher can also isolate objects to create sets and then ask the children to identify the rule of the set: “What’s my rule?” (see image above). This is more complex than matching since, while the objects are all the same shape, they are not all the same size or colour. The children can also be encouraged to play the “What’s my rule?” game in groups.
  • Initially, the isolated objects should only have one attribute in common, e.g. in the image above, there is the set of all the shapes that are square and all the shapes that are not squares.
  • Ultimately, it is hoped that the children appreciate that while the collection above has been classified according to  a certain attribute (i.e. whether it is a square or not a square), that the same collection can be sorted in various other ways  e.g. triangles/not triangles; pink/not pink; big/not big. And then for these children, they can be asked to identify the rule of a set that have two attributes in common e.g. a set of yellow squares (which then also creates by default a set of shapes that are not yellow squares).
  • The children will likely begin themselves to sort objects into multiple sets. Instead of two sets of yellow shapes and non-yellow shapes (i.e. a binary sort) they will produce a set of yellow shapes, red shapes, blue shapes etc. The production of multiple sets will naturally lead on to comparing and ordering these sets (see next section).

HINT: The children themselves can also be used for classifying; use rope or yarn circles on the ground, and ask a small group of children up to stand at the top of the class. Point to each of the sets saying “This is for the children wearing glasses and this is for the children not wearing glasses”; “This is for the children with curley hair and this is for the children who don’t have curly hair”; “This is for the children with brown eyes and this is for the children who don’t have brown eyes” etc. 

Comparing and Ordering

Comparing is instinctive in humans, and children are no exception to this: “He got more than me! I have a smaller piece!”

Comparing is also intrinsically connected with matching and classifying: when a child explains that two shapes are different/not the same because one is yellow and one is red, they are already comparing according to colour. When a child classifies a set into big toys and small toys, they are already comparing the items in the sets according to size. Therefore, to compare is to measure or quantify in some way how two items or two sets are similar or different.

As well as colour and size, the children can also compare objects according to length, width, height, weight or thickness.

  • Use collections of pencils, crayons, ribbons, strings etc., to compare length. Note how the children do this; do they use a common baseline/starting point, and if not, highlight the need for the same.
  • Sort attribute shapes into sets that are thick and thin
  • Use the opportunities to introduce vocabulary that will reinforced later in the year as part of measures e.g. long/short, longer/shorter, heavy/light etc.
  • Compare sets without counting: when sorting, look for opportunities where the resulting sets are obviously different in quantity and ask the children to identify which has more and which has less. Some children may even demonstrate their ability to verify their comparison by counting; this is an added bonus, but not required from all the class at this stage.

Ordering is a development of comparing, in that the children are now comparing three or more objects and ordering them according to length, height etc. The children can compare and then order sets also; “there are more yellow shapes than red shapes, there are more blue shapes than yellow shapes, (then in desceneding order) so it’s blue shapes, then yellow shapes, then red shapes”. An important conceptual development is where the child realises that if A is more/bigger than B and B is more/bigger than C, then A has to be more/bigger than C, and thus A must be the largest and C must be the smallest.

HINT: In your materials for EMA include (real or play) coins and notes, with the  emphasis being on their attributes of material, colour, shape, size and design. Provide the children with opportunities to suggest ways to classify the coins themselves. Using money in this way is an excellent way to prepare them for the strand units of money, later in the year.  NB: While the emphasis should be on the attributes of the coins/notes, as opposed to their value and/or the numbers visible on them,  if children recognise their value and use as an attribute for matching, classifying, comparing and ordering the coins/notes, then this should be acknowledged as a valid response to the activity.

Further Reading and Resources:

This is part of the series “Digging Deeper into …” which takes 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.

Digging Deeper into … Number Sentences, Equations and Variables (3rd – 6th)

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For practical suggestions for families, and helpful links to digital resources, to support children learning about this topic, please check out the following post: Dear Family, your Operation Maths Guide to Number Sentences and Equations

In the Primary Mathematics Curriculum (1999), this topic appears as three separate strand units, all within the strand of Algebra:

  • Number Sentences (3rd & 4th class)
  • Equations (5th & 6th class)
  • Variables (6th class)

However, since these concepts are intrinsically connected, in Operation Maths they are taught in a cohesive and progressive way through third to sixth class.

  • Number sentences are mathematical sentences written using numerals (e.g. 1, 5, 67, 809, 1.45, 1/2  etc.) and mathematical symbols (e.g. +, -, x, ÷, <, >, =).
    • They include both equations (see below) and inequalities (64 < 82, 23 > -16), although the term inequalities is not specifically used.
    • The unknown or missing value in a number sentence (i.e. a variable) can be represented by a frame (box), by a shape, or by a letter, although it should be noted that the Primary Mathematics Curriculum (1999) specifies a preference for a frame (box), up to the introduction of variables in 6th class
  • An Equation is a special type of number sentence, containing an equals sign, to show that two expressions are equal (e.g. 5 = 3 + 2, 5 + 6 = 20 – 9, etc.)
  • A variable is a value in an expression that can  change or vary. However, when there is only one variable in an equation then the value of that variable can be calculated e.g. a + 6 = 9, 20 = 4b.

Thus, while these strand units are only being formally introduced from third class on, the children have actually been exposed to number sentences, equations and variables (i.e. the frame) since the infant classes.

 

Equations

(aka Number sentences with an equals sign)

Understanding equations necessitate the appreciation of the correct meaning of the equals symbol. Many children incorrectly translate the equals symbol (=) as meaning ‘and the answer is…’. This incorrectly reinforces that both its purpose and position is to precede the answer in any calculation, a misconception also reinforced by calculators, where you press the = button to get the answer. Such misunderstanding is
evident when you see responses like these:
5 + 6 = [11] + 3 , i.e. ‘5 + 6 is 11’
5 + 6 = [14] + 3 , i.e. ‘5 + 6 + 3 is 14’
Adults may also unwittingly compound this, by using ‘makes’ or ‘gives’ as a synonym for equals.

It is vital that the children recognise that the equals symbol indicates that both sides of the equation (which will be referred to simply as a number sentence until fifth class, when the term “equation” is introduced), are equal to one another/are the same value/are balanced. In this way an actual balance (pan or bucket) and cubes can be extremely valuable to model (and solve) equations e.g. in the images below, the first balance shows that 5 equals a group of 3 and a group of 2, and the second balance shows that 12 equals 3 groups of 4.

From Operation Maths 4

From Operation Maths 5

Furthermore, teachers should reinforce the correct meaning for the symbol = by only translating it as ‘equals’, ‘is equal to’ and/or ‘is the same as’.

 

Inequalities

(aka Number sentences with greater than/less than sign)

Despite the fact that the children have been using the greater than and less than symbols since 2nd Class, many still have difficulties reading them and interpreting their meaning. Using a balance and concrete materials, in a similar way as when teaching equations, can greatly help children to gain deeper understanding of the symbols and their meanings.

From Operation Maths 4

Through exploration they can identify what is the maximum number of cubes they can put on a side that is less than the other side, before it makes the balance tip in the other direction, thereby invalidating the number sentence; or the minimum number of cubes they can put on a side that is greater than the opposite side, so as to keep the number sentence true.

 

Using estimation strategies

Often, when having to indicate if a given number sentence is true or false, it is not always necessary for the children to calculate both sides of the number sentence exactly. There is (usually) only one true or correct option, meaning that every other answer is incorrect or false. Encourage the children to use their estimation and number sense skills to quickly recognise when a statement is obviously false, e.g. a big difference in the size of numbers on one side versus the other.

While some might view this as a type of ‘cheat’ strategy, in truth, it is more about identifying the most efficient approach, while also reinforcing the value of estimation in general and, particularly, as a way to make calculations easier.

 

Translating number sentences into word problems and vice-versa

As mentioned earlier, this is in fact a skill that the children would have been exposed to, and been using, since the infant classes. Furthermore, as this topic is deliberately positioned towards the end of the yearly plans in Operation Maths 3-6, the children will have already been using this skill very regularly in the number, data and measures chapters, prior to this point of the school year.

The curriculum specifies that the children should be enabled to translate number sentences into word problems, both of which can be viewed as abstract representations. Worth noting, is that the curriculum doesn’t emphasise the importance of the translating the number sentences and word problems into concrete and/or pictorial representations. Whereas, in Operation Maths, (in keeping with its overarching CPA approach) , there is significant emphasis placed also on utilising various concrete materials and visual strategies to represent the word problems and number sentences.

From Operation Maths 5

The development of visual strategies for problem-solving,  is a central focus of the work throughout the Number chapters. Thus, this topic allows the teacher to revise the visual strategies covered so far and assess how well the children understand them and can apply them.

The interconnectedness of real-life scenarios and mathematical sentences/equations should also be emphasised. At primary level, there should always be some relatable context for any number sentence.
For many children, when looking at a number sentence, it can be difficult to appreciate how a collection of digits and symbols could relate to a real-life scenario with which they can identify. That is essentially what a word problem is; it provides a real-life context within which to frame the numbers and operators involved. Emphasise to the children throughout this topic how the number sentences could be given a real-life story (i.e. word problem), and encourage them to come up with possible stories either verbally or written down.

And, depending on the context given to a particular story, the visual representation may also be different, even though the number sentence/equation may stay the same. For example for the number sentence 7 – 4 = ? the word problem (context) could be either of the following:

  • Ali had 7 cookies. He ate 4 cookies. How many cookies does he have left?
  • Áine has 7 cookies. Abdul has 4 cookies. How many more cookies has Áine than Abdul?

Image created using Bar Modelling eManipulative, accessible on edcolearning.ie

And while the number sentences are the same, both the contexts and the pictorial representations (e.g. bar models, as shown above) are different, as they represent different types of subtraction. In the case of the first word problem, this is describing subtraction as deduction, and a part-whole bar model is more suitable. In the case of the second word problem, this is describing subtraction as difference, and a comparison bar model is more suitable.

 

Identifying operation phrases

When the children are translating word problems into number sentences, it is very important that they can understand the context being described and are able to identify that phrases that indicate the operation(s) required.

Regularly interspersed throughout the operations chapters in the Discovery books for Operation Maths 3-6,  there are activities which enable the children to identify and colour-code the specific vocabulary that an indicate the required operation (see example below). This topic provides an ideal opportunity to review this skill and assess/re-teach the children accordingly.

From Operation Maths 4, Discovery Book

In particular, many of the Talk Time activities, require the children to suggest ways to verbalise the various equations, e.g:

  • The difference between 46 and 18 is equal to the product of 4 and 7; true or false?
  • 18 subtracted from 46 equals 4 times 7; true or false?

Where possible the children should suggest alternative phrases for the same equation thus reinforcing the use of correct mathematical language.

 

Input and Outputs

In Operation Maths 4 & 5, activities based on inputs and outputs are included as a means to consolidate the children’s understanding of number sentences and their ability to write number sentences (see below).  Input-output activities can provide great scope for problem solving, as well as preparing the children for calculations involving variables in sixth class.

From Operation Maths 4

 

Variables

Variables are formally introduced in sixth class, although the children have encountered variables (as a symbol or shape to represent a missing value) since they first encountered the frame (answer box).

When calculating with variables, both part-whole bar models and comparison bar models can be very useful to represent the relationship between the known and unknown values.

From Operation Maths 6

Further Reading and Resources

Digging Deeper into … Capacity (all classes)

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For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of capacity, please check out the following post: Dear Family, your Operation Maths Guide to Capacity

Strictly speaking, capacity is the amount (or measure) of a substance (which can be solid, liquid or gas) that something can hold (i.e. a container). That said, in primary mathematics we tend to use capacity as a measure of liquids only (ie not solids or gases), both to avoid confusion and since the children would most commonly see examples of liquids measured using the standard units of capacity (ie litres and millilitres).

Initial exploration – CPA approach

Like the topics of Length and Weight, and in keeping with the over-arching CPA approach of Operation Maths, children’s initial experiences of capacity at every class level should focus on hands-on activities, using appropriate concrete materials.

In the younger classes, this should occur through exploration, discussion, and use of appropriate vocabulary eg full, nearly full, empty, holds more, holds less, holds as much as/the same as etc. The children should also be enabled to sort, compare and order containers according to capacity.

From Operation Maths 1

 

Irrespective of the class level, introductory exploration in this topic could follow the following progression or similar:

  • The children examine pairs of empty containers and make comparisons, so as to identify, from sight, which holds more/less. Use questioning to encourage them to assess all available information:
    • Which container is wider/narrower?
    • Which container is taller/shorter?
  • Elicit from the children how they might verify their estimates. Introduce a non-standard measure (e.g. egg-cup, yogurt container, plastic cap from an aerosol, tea/table spoons, plastic syringe, flask etc) and demonstrate how to measure the capacity of a container using a non-standard measure eg (using egg-cup as standard measure):
    • Fill an egg-cup with water. Pour this into the target container to be measured. Repeat until container is full and then record the number of egg-cups required.
    • OR fill the container with water. From this, pour out an egg-cup full, which is then poured out into a third container (eg basin, plastic box). Repeat until the target container is empty and then record the number of egg-cups that were filled from it.
    • OR fill the target container with water. Pour this into a larger container and record the level of the water by marking the level on the side. Pour out the water out into a third container (eg basin, plastic box) to be used as a water store/reservoir. Repeat with other containers to be measured and use the marking on the side of the measuring container to identify which container held the most/least etc. Please note though, that while this method can be used to identify which container holds the most/least, it will not provide a measure of the capacity as a quantity of  non-standard units (unless of course the measuring container has existing markings for litres and/or millilitres)

 

 

From Operation Maths 1

 

HINT: In order to be avoid unnecessary water wastage and/or a very wet classroom (!), it can be a good idea to conduct the capacity activities outside and over a number of plastic basins/boxes. These can be used to catch spills and to hold the water which can be re-used repeatedly to measure the capacity of the various containers. 20 ml or 50 ml plastic syringes can also be very useful; they are easy for smaller hands to use draw up water and squirt it into a container. And instead of counting ml, ask the children just to record the capacity of the container as the number (count) of syringes that it can hold.

Move on to pairs of containers whose difference in capacity may not be obvious because of the shape and dimension of the containers. Thus, it is important to use a selection of containers that vary in height and width.

This can then progress to incorporate a direct comparison of the capacity of three or more containers. It is important at this stage that the children realise that if A holds more than B and B holds more than C, then, without further direct comparisons, we know that A holds more than C, that A holds the most of all three and C holds the least. This is a very important concept for the children to grasp.

HINT: Use brainstorming to elicit the names of various liquids and container types with which the children are familiar. Use the list to make up an odd one out game, as outlined below

From Operation Maths 2 TRB (similar activity also in Operation Maths 1 TRB)

  • In a similar way, the children can estimate and record the capacity of containers of objects using standard units (i.e. litres and millilitres; the latter is introduced in third class). Initially, when using the standard unit of a litre (starting from first class) the children will be recording the capacity of containers as being able to hold more than/less than/the same as a litre.

HINT: In 2nd class & 3rd class the children will be using 1/2 litre and 1/4 litre (as opposed to millilitres). This will necessitate using bottles etc that are marked in 1/4 litre intervals. Challenge the children in these classes up to come up with ways to measure and mark these intervals, without having to use millilitres or some type of commercial graduated measure (eg a jug). This task could be given as an alternative homework activity.

When finding the capacity of a container, it is important also to highlight to the children that it is not necessary to fill it to the brim. Show them an example of an unopened litre bottle of water – the height of the water in the
bottle is not to the brim, yet the label shows it contains 1 litre. Thus, the children will develop an understanding that the actual capacity of containers are typically greater than the indicated capacity of the liquid it contains.

Problem Solving: How many are needed to fill? It takes 4 of container A to fill container B. It takes 2 of container B to fill container C. How many of container A are needed to fill C? This can be a very difficult concept to grasp for many children. Some suggestions include using multiples of the real containers to show the relationships between each and drawing pictorial representations using bar models, one of the three key visual strategies for problem-solving used throughout Operation Maths, (shown below). 

Using more accurate measures

As the children progress in their understanding of the concept of capacity they will begin to appreciate the need for more accurate means to record it; both using smaller standard units (ie millilites) and using measures/containers which are already calibrated/graduated with markings. It is an advantage to have a wide selection of different types of measuring instruments available (including plastic jugs, syringes, measuring spoons, graduated cylinders etc) so that the children appreciate that different measuring instruments are more suitable for certain tasks. When measuring, advise the children also to read the level of liquid at eye level to obtain a more accurate reading.

HINT: Some jugs etc can be purchased relatively cheaply from value shops. Alternatively, ask the children to bring in measuring jugs, containers etc., from home to use in class while working on this topic.

As always, the children should be encouraged to estimate before measuring.  And, rather than estimating the capacity of A, B, C and D before measuring A, B, C and D, it would be better if the children estimated the capacity of A and then measured the capacity of A, estimated the capacity of B and then measured the capacity of B and so on. Thus, they can reflect on the reasonableness of their original estimate each time and use this to refine their next estimate so that it might be more accurate. This helps them internalise a sense of capacity, and to use this sense to produce more accurate estimates.

When the children have experienced using a variety of instruments for measuring capacity, they should then be afforded the opportunity to choose which instrument (and which standard unit) is most appropriate to measure the capacity of various containers. In this way, the children start developing the notion that while many approaches can be taken, some are more efficient than others, and the most efficient approach will also depend on the target object being measured. This is the same as the Operation Maths approach to operations throughout the classes; there can be many approaches and some are more efficient than others, depending on the numbers/operations involved.  The aim is for the children to become accurate, efficient and flexible thinkers.

Renaming units of capacity

From fourth class on, the children will be expected to rename units of capacity, appropriate to their class level. While changing 1,250 ml to 1 l 250ml or 1.25 l, will typically be done correctly, converting figures which require zero as a placeholder (eg 1 l 50 ml, 2.6 l ) can be more problematic, and can reveal an underlying gap in understanding, that is not revealed by the more obvious measures. In these cases, the children should be encouraged to return to the concrete experiences as a way of checking the reasonableness of their answers, eg:

  • “1 l 5o ml…well 1 l  is 1,000 ml and then there’s 50 ml more so it’s 1,000 plus 50, which is 1,050 ml.
  • “2.6 l equals 2,600 ml because 1 l is 1,000 ml, so  2 l is 2,000 ml and .6 is slightly more than .5, which is half of a l or 500 ml, which means .6 must be 600 ml”

T-charts, another one of the three key visual strategies for problem-solving used throughout Operation Maths, can be very useful when renaming units of capacity, as can be seen below. These can be partially started on a class board and the children then asked to complete the T-chart with their own choice of capacities as is relevant to the tasks required of them. The children could construct these also to use as a reference, as they progress through this topic.

 

 

Capacity & Volume

Volume is introduced officially for the first time in 6th class. It is preferable to introduce the children to volume via cubed units (eg blocks) as opposed to via cubed centimetres (see below).

From Operation Maths 6

 

HINT: Did you know that the smallest base-ten blocks (ie those often used as units or thousandths),  are 1 cm cubed? This means that these could be used to build shapes from which the volume of the shapes can be measured and they can be used to measure the approximate volume of an open cuboid eg lunch box, pencil box, etc.

The children may find it challenging to appreciate the relationship between capacity and volume, especially since they may think capacity is exclusive to liquids while volume relates to solids. Providing the children with opportunities to measure the the capacity of a variety of different sized cuboids (eg lunch box) and then measuring its volume using 1 cm cubes, will likely lead the children to discover the connection between the two concepts and that 1cm cubed equals 1 ml.

From Operation Maths 6



Further Reading and Resources:

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