Author Archives: Operation Maths

Digging Deeper into … Fractions (1st – 6th)

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

‘The headlong rush into computation with fractions, using such mumbo-jumbo as “add the tops but not the bottoms” or “turn it upside-down and multiply”, has often been attempted before the idea of a fraction or fractional notation has been fully understood.’

Nuffield Maths 3 Teachers’ Handbook, Longman 1991

Fractions, as is often acknowledged, can be a very problematic topic; so much of this concept appears to be at odds with key concepts in other areas, eg:

  • Children’s understanding of number tells them that 4 is greater than 2, yet 1/4 is not greater than 1/2 of the same shape. Similarly, fractions cannot be ordered using whole number understanding.
  • The size/value of a fraction depends on the size/value of the whole from which the fraction is taken, i.e. a quarter of one number, e.g. 100, can be greater than a half of another number, e.g. 30. From the beginning, it’s important to stress that fractions are not whole numbers, but rather, they represent bits or parts of a whole and that the whole can be different each time.
  • Equivalent fractions of the same shape do not always have to be the same shape (congruent), as can be seen from the quarters of the square in the example opposite, but each quarter should have the same area.
  • Fractional language and terminology can pose difficulties. Verbally there is little to distinguish between eighth and eight, ten and tenth, etc. In addition, eighth can be one whole cut into eight equal pieces, and as an ordinal number it represents the position between seventh and ninth.

Thus, not only is it very important that the children appreciate the importance of accuracy when communicating about fractions, it is essential that children are given sufficient opportunities to develop a solid understanding of fractions: what they are, how they are made, changed, used, etc. Therefore, as typical in Operation Maths, the activities in this topic follow a concrete–pictorial–abstract approach.

 

Variety of representations

In Operation Maths, a variety of activity types are incorporated, based on linear, area (shapes) and set models. It is very important that children use a variety of materials and representations when exploring fractions, as otherwise they may only be able to relate their fractional understanding to a certain model.

  • Area models: includes dividing shapes such as circles, squares, rectangles etc into various fractions and/or using these type of representations.
  • Linear models: includes using strips of cubes to show fractions, marking fractions on a number line, comparing fractions on a fraction wall, using bar models
  • Set models: Dividing a set of identical objects or a number into halves, quarters etc; identifying or calculating the whole amount when a fraction of the whole is known etc.

For more information on the various types of fractions models watch any of the videos at the links below:

In keeping with the overall CPA approach of Operation Maths, the initial part of the fraction chapters will focus on the children having plenty of experiences using concrete materials (e.g. creating fractions using the various models, comparing and ordering them) and/or interacting with pictorial representations (e.g. combining and partitioning fractions and labelling them using the appropriate names and symbols). It is important that the children have sufficient experiences at these levels before progressing to more abstract tasks e.g. calculating with fractions.

In the Operation Maths TRBs there are a multitude of suggested tasks, including fraction stations, based on the exploration and use of concrete manipulatives. Other suitable concrete-based activities include using Lego and/or pattern blocks to explore fractions.

Problem solving strategies that have been used previously to promote visualisation, should also be incorporated into this topic, including bar models, empty number lines and T-charts (see images below).

 

 

 

Calculating fractions

When calculating a fraction of an amount, or calculating the whole amount given a fraction, bar models can be used to represent the information and solve for the answer (see bar models above). The children should also be encouraged to develop and use strategies to calculate mentally where possible, for example partition the amount into friendly numbers (using branching, if written, as shown below) and then re-combine.

 

Operations with fractions (5th & 6th)

Fifth class is the first time that children will formally encounter operations involving fractions. The curriculum for fifth class requires that the children be enabled to ‘add and subtract simple fractions and simple mixed numbers’ and to ‘multiply a fraction by a whole number’. The emphasis here should be on the word simple, and the simplest fractions for most children are likely to be halves, quarters and eighths, since these are the fractions the children are most likely to encounter in real life, as well as being among the earliest fractions they met in primary mathematics. Therefore, it is sufficient for some children to be enabled to achieve this objective using only wholes, halves, quarters and eighths. Other more-able children should be allowed to extend their understanding to other fractions, e.g. fifths and tenths, then thirds, sixths, ninths and twelfths, and finally to unconnected fractions if relevant, such as sevenths and elevenths.

Strategies that are used in Operation Maths for addition and subtraction of whole numbers, (and that are used in Number Talks) can often be applicable for addition and subtraction using fractions also e.g.:

  • Compensation: moving an amount from one addend to the other addend to make a more friendly (compatible) number e.g. in example A above, moving 6/10 to 2/5 to make 1 whole unit.
  • Partitioning: separating the whole numbers from the fractions and adding each separately, before recombining, as shown in example B above.

Similarly for subtraction, fraction number lines can be used to model either the taking away (deduction) strategy or the difference (adding up) strategy. A selection of number lines for this very purpose are provided on the inside back cover of the Operation Maths pupils books for 5th and 6th classes.

 

Ratios

Sixth class is the first time that children will formally encounter ratios. It is important that they realise that ratios don’t necessarily tell you the quantity of items in a group/set, rather how many of one quantity there is to an amount of another quantity. For example the ratio of teachers to pupils in the school is 1:27. That doesn’t tell us that there is only one teacher or just 27 students in the school, rather that there is a teacher for every 27 students.

 

Misleading rules

Avoid relying on rules such as ‘divide by the bottom, multiply by the top’, ‘add/subtract the tops but not the bottoms’, ‘multiply tops and bottoms‘. These rules are very abstract and do not encourage the children to visualise the numbers or fractions involved and what is being asked. Such rules also reinforce incorrectly that teaching maths is about teaching the children to do maths, rather than teaching them to understand maths. And, even if such rules are used to calculate correctly, it can often be in the wrong situation, e.g. ‘What is the whole number if 3/4 is 12’, to which the child “divides by the bottom, multiplies by the top” to give an incorrect answer of 9.

Rather, the emphasis should be on the children developing the ability to visualise fractions and fractions of sets/numbers. Using concrete materials and pictorial representations (e.g. fraction pie pieces, number lines, arrays, bar model drawings, etc.) can greatly aid this. Ultimately, it is hoped that the children will identify some shortcuts themselves (e.g. simplifying the multiplication of fractions), that may echo the traditional fraction rules; however, the emphasis needs to be on these arising from the children’s own discoveries.

 

Further Reading and Resources

 

 

Digging Deeper into … Money (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 money, please check out the following post: Dear Family, your Operation Maths Guide to Money

Similar to the strand unit of time, money is an integral element of our daily lives and therefore an essential, “need-to-know” topic in primary maths, particularly for those children with special educational needs or learning difficulties. And, while it is not as abstract as the topic of time, it still can be a concept with which many children struggle. Consider the nature of money itself:

  • It comes in different shapes and sizes, and in metal and paper forms (i.e. coins and notes) each of which has its own value.
  • The sizes of the coins and notes are not proportional to their value i.e. a 20c coin is not twice as big as a 10c coin; a €100 is not ten times the size of the €10. Therefore, while money can be used as a base-ten material, unlike the base-ten blocks, it is not proportional.
  • Money can be expressed as € or c, but not as both. And when using the € sign it precedes the numbers (even though it is verbalised as “six euro” as opposed to “euro six”), where as the c sign comes after the numeral.
  • Countries often use different currencies; this can lead to confusion when children presume that dollars and pounds are used in this country, because they hear this terminology regularly from imported TV programmes.
  • When changing currencies you cannot do a straight swap i.e. €1 doesn’t equal £1 or $1; the new value must be calculated using an exchange rate, which also varies.
  • More and more, transactions are becoming cashless, as people use credit and debit cards more than ever before. Thus, children are missing out on essential opportunities to handle cash, or see it being handled in real-life situations. The increased use of plastic and contactless payments also limits the opportunities for people to total mentally, calculate change etc.

 

Elicit prior knowledge & concrete exploration

At every class level, it is always a good idea to elicit the children’s prior knowledge, which can be very varied, depending on the experiences they’ve had with money. Even some simple revision questions can be very revealing, such as these:

  • ‘What currency/money do we use in Ireland?’
  • ‘Do you know of any other countries that use the euro?’
  • ‘Can you name the coin/note with the least value? And the next? And the next?’ etc

Even in a senior class, the answers to the last question can often be ‘1c, 2c, 3c, 4c…’ as the children forget or don’t realise that there is not a single coin for each value. In keeping with the CPA approach of Operation Maths,  these type of questions should be followed with opportunities to explore and examine the actual notes and coins, and the similarities and differences between them. And, if there is not enough real or replica money, the Sorting eManipulative, accessible via edcolearning.ie, can be a very useful way to display the coins and notes (see below).

Do you notice any pattern? Many children, and even some adults, don’t recognise that our euro money follows a ‘1, 2, 5’ pattern i.e. every note or coin has either 1, 2 or 5 as its most significant digit (look at the columns above). Once the children recognise this, they are less likely to suggest using a ‘3c coin’ or a ‘7c coin’ etc to make a value. To improve their familiarity with the coins, even children in junior infants could use them for pre-number sorting purposes, eg using coins for sorting by size, colour, shape etc. They don’t need to be restricted to just coins up to 5c (the traditional limit for junior infants), as the focus is not on number. Play activities based on money e.g. the shop, post office, restaurant etc should also be encouraged, particularly as the basis for Aistear themes.

 

Exploring the value of the coins

Recognising the value of each individual coin is one thing, recognising that one coin can be exchanged for a number of coins of equal value is very different. This is why it can be very useful to represent the value of the coins concretely. This can be done by attaching coins to large squared card and/or ten and five frames. Grid paper with 2cm squares is perfect for this. Just print out/photocopy onto white or coloured paper or light card and then cut out into sections that relate to five and ten frames (see image below):

  • 1c, 2c, 5c on to strips of 1, 2 or 5 squares respectively
  • 10c onto a 2×5 section i.e. ten frame
  • 20c onto a 4×5 section, with a bold line through centre to show each ten
  • 50c onto a 10×5 section, with 4 bold lines to show each ten

This reinforces the benchmarks of five and ten, while building on the children’s ability to subitise (recognise at a glance) these quantities.

These materials can then be used for exchanging activities, where the children identify different ways to make various values, i.e “same value, different appearance”,  e.g. what coins could we use instead of 2c, 5c, 10c, 20c, 50c, etc. When the children are comfortable making these values they should then make values that are not equivalent to a single coin e.g. 6c, 13c, 23c etc. Ultimately, it is hoped that the children will be able to visualise the value of the coins without needing the visual supports shown above.

 

Mental calculations with money

Despite the face that our society is becoming increasingly cashless, mental calculations with money should still be emphasised and, in particular, the strategy of making change. Officially referred to a complementary addition, where you add on to subtract, it has also been known as “shopkeepers arithmetic” given its application in those situations. It is also one of the specific subtraction strategies dealt with in Number Talks, where it is referred to as Adding up (all of the other Number Talks strategies are also relevant to calculations involving money, but this one is worthy of a special mention).  Complementary addition is also one of the strategies used in Operation Maths, where it is shown using the visual strategy of an empty number line (see below).

 

Visual Strategies for Problem Solving

A key element of Operation Maths is the use of three specific visual strategies to support the development of problem solving skills. These are empty number lines (as shown above), bar models and T-charts. T-charts are particularly useful to solve problems based on the unitary method, as shown below.

Bar models can be very useful to model addition and subtraction problems e.g. where the whole amount is known and a part is missing or where the parts are known and the whole is missing. The type of models shown below are referred to a part-whole models.

For more information on the visual problem solving strategies used in Operation Maths 3-6, please read this post.

 

Other tips and suggestions for teaching money

  • Emphasise that money is based on the euro. Cent coins are merely fractions of that unit; euro coins and notes are multiples of that unit. In this way money can also be used to help teach other strand units, including place value, operations and decimals (see examples below).
  • Using money to represent place value with whole numbers and decimals
    Using money to model 2-digit division
    Using money to model decimals to hundredths
  • Emphasise the importance of using an efficient estimation strategy when calculating with money:
    • front-end estimation: where you only consider the most significant digit i.e. think of €26.95 as €20
    • rounding: where you round to the nearest unit, ten etc, i.e. think of €26.95 as €27 or as €30. Rounding produces more accurate estimates than front-end estimation; however, it can also be more time-consuming for some children and thus less efficient.
  • Value for money: Encourage the children to compare prices in different shops and/or catalogues to identify the best price and when comparing items being sold as multiples to compare their values using the unitary method and T-charts, as shown above.

Further Reading and Resources:

Digging Deeper into … 3-D Objects

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For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 3-D objects, please check out the following post: Dear Family, your Operation Maths Guide to 3-D Objects

The first obvious question to be addressed is why this topic is called 3-D objects in Operation Maths, when most other textbooks, and even the curriculum refer to this topic as 3-D shapes?

In the PDST Shape and Space manual, it is suggested that “using the word ‘shape’ to describe 2-D shapes and 3-D shapes can cause confusion for pupils”. For example, asking pupils to ‘describe the shape of this shape’ highlights one problem. Another problem is that pupils must be able to think of all cuboids as being ‘the same shape’, while mathematically speaking all cuboids are not the same shape.

The manual goes on to suggest that it would be more helpful to refer to 3-D things as ‘objects’. Using ‘objects’ also reinforces the notion that if it can be physically handled/picked up, it must be a 3-D object, as opposed to a 2-D shape which should always only have length and width, not depth/height.

So, throughout the Operation Maths books, this topic is called 3-D objects to avoid confusion and to provide clarity for the pupils. However, wherever there is reference to “strand unit”, the term 3-D shapes is used, as this is the term used in the curriculum.

CPA

As usual, this topic is explored using a CPA approach, where the initial focus is on the exploration of 3-D objects and the identification of similarly shapes objects from the child’s environments. And, in a similar way to the teaching of 2D shapes, while the 3-D objects will be identified by name, the greater focus should be on their properties, as appropriate to each class level e.g.

  • identifying whether the objects roll, stack, or slide
  • relating the properties of each object to its purpose and use in the environment
  • recording, sorting and comparing according to the number and shape of faces and curved surfaces
  • recording, sorting and comparing according to the number of edges and vertices (corners)

It is important that the children discover the properties of the 3-D objects by hands-on investigations and by classifying and sorting. Sorting activities help develop the children’s communication, observation, reasoning and categorising skills, and thus will help to develop a conceptual understanding of the objects. In the senior classes using online interactive sorting activities based on Venn diagrams or Carroll diagrams can be very useful. Asking children to bring in examples of 3-D objects from home will help them to become aware of 3-D objects in their environment.

Faces, curved surfaces and edges

How many faces on a cylinder? Three or two?

Traditionally, in Ireland, and in Irish textbooks, a cylinder was recorded as having three faces. However, this is not mathematically correct, as strictly speaking a face is flat, and a 2D shape (figure), so therefore a cylinder has in fact only two faces, (both circles), and one curved surface. And while some may argue that a cylinder has a third face i.e. the rectangular shape you see when you disassemble the net of the 3-D object, in this disassembled state it is no longer a cylinder, since it can no longer roll, a specific property of all cylinders.

Another way to think about the faces of 3-D objects is to consider the number and shape of the resulting outlines of tracing around each surface of the 3-D object. It is only possible to trace around the opposite ends/bases of the cylinder, since only these are flat, thus it has only two faces, both of which are circular in shape. Similarly, it is only possible to trace around one surface on a cone, which therefore means it has only one face (a circle) and one curved surface.

And how many edges on a cylinder? Officially none, as an edge is where two flat faces meet and the faces on a cylinder are on opposite sides and do not touch/meet. However, that leaves the problem of how to describe the place where each face meets the curved surface.  So in Operation Maths, as occurs typically in other primary texts in other countries, there is a distinction made between straight edges (which are in fact true edges) and curved edges (which strictly speaking are not edges).

In 5th and 6th class, it is important that the children realise that only some of the 3D objects are also polyhedra, and that only five of these are categorised as platonic solids.

Further Reading and Resources

Digging deeper into … the Circle (5th & 6th)

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While the circle, as a topic, is not a specific strand unit in itself, it traditionally has been dealt with separately, and more in depth, in 5th and 6th class. It is in these classes that the children begin to explore the circle as something more than just a 2D shape, and rather as a shape which has specific parts that can be named (eg diameter, radius, etc) and specific properties that can be explored. Fifth class is also the first time that the children have encountered degrees as a measurement of rotation (they first meet degrees in the strand unit of lines and angles). Fifth class is also the first occasion when the children would be required to use degrees to construct pie-charts in the strand unit of representing and interpreting data, for which the children require knowledge of constructing circles and dividing them into sectors prior to creating pie charts.

Thus, in Operation Maths, this chapter is placed after the chapters of Lines and Angles and 2-D Shapes and before the chapter of Data.

CPA Approach

As is typical in Operation Maths, A CPA approach is taken to this topic where the emphasis at the introductory stages is on the children exploring and examining circles in their environment and then using the manipulation of cut-out circles to identify the parts of circle and label them using the correct mathematical terminology. Indeed the children themselves can be used to model a circle, as outlined below. This type of activity will also greatly suit the kinaesthetic learners.

When looking for circles in the environment, it is also important to accurately identify circles, which are examples of 2D shapes, as distinct from cylinders, which are examples of 3D objects. So, if a child suggests that a coin is an example of a circle, emphasise that the face of a coin is indeed a circle, but that the coin itself is a cylinder.

Integrating maths and literacy

When introducing this new terminology, ask the children to suggest examples of words with similar prefixes/rootwords so as to foster connections and deepen meaning eg diameter coming from dia meaning across, through and thus connected with diagonal, diaphragm, dialogue; radius as being related to radiate, radio, radar (originating from a central point and moving outwards), etc. Etymology websites, such as Etymonline can be very useful to research and collect related words.

Measuring and constructing circles

The children should be provided with ample opportunities to measure the radius and diameter of circles of various sizes and, in doing so, be guided to discover for themselves that the measure of the diameter is twice the radius.

In a similar way, in 6th class, through comparing the measurement of the circumference and diameter of various circles, it is hoped that the children realise that the circumference of a circle is always just over three times the measure of the diameter. They can explore this in a very concrete way by measuring the circumference of various circles using string/wool and then cutting the string into lengths that equal the diameter, as show in the image below from K-5 Math Teaching Resources

From K-5 Math Teaching Resources

Again before constructing circles using a compass, the children should be asked to suggest ways to draw circles that they may have used previously and to identify the pros and cons of these methods. When ready to use a compass, it can be a good idea to use a video to demonstrate, such as the ones below.

 

Circles in art

Once the children have mastered the basics of constructing a circle, they could be encouraged to look and respond to circles in art, for example the work of Kandinsky, Anwar Jalal Shemza or the multitude of circle themed pieces available to view on the internet (just search google images for circles in art). The children could even look and respond to crop circles (in particular the activities of John Lundberg) or the use of circles in famous brand logos. For more ideas, check out this Pinterest board which includes circle themed art lessons. Using circles in such a way provides purposeful opportunities to use and reinforce the specific circle terminology e.g. diameter, circumference, arc, etc.

Further Reading:

Shape and Space Manual from PDST, p.154-157

Digging Deeper into … Number Theory (5th & 6th classes)

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

Number Theory is a number topic that is concerned with subcategories of whole numbers: odd and even; factors and multiples; prime and composite; square, triangular and rectangular numbers; square roots and exponential numbers.

While 5th class is the first time that the children formally meet this strand unit, they have engaged with elements of the topic in previous classes:

  • odd and even numbers were explored formally in 1st, 2nd and 3rd classes
  • the language of factors and multiples was used in 3rd, 4th and 5th classes as part of the multiplication and division chapters.

Concrete-pictorial-abstract approach (CPA)

Traditionally this topic is often taught in a quite abstract way e.g. using operations to calculate square numbers, square roots identifying prime numbers by the number of  factors, etc. However, number theory by its nature, prompts us to consider and explore the shape and arrangements of numbers and, thus, is ideal to be explored in a visual way. Even Lego can be used to explore square numbers, arrays and factors!

Therefore, the activities will once again follow a concrete-pictorial-abstract approach, as is used throughout Operation Maths with the emphasis being on the children being able to build and/or visualise the shape of the numbers themselves, before progressing to using more abstract means to identify larger numbers, compare patterns, etc.

Therefore, the focus in this topic should not be on the numerals themselves, rather on the shape of the numbers; it is one thing to recognise or identify odd and even numbers, factors, prime, composite, squared numbers, cubed numbers, etc., it’s another thing to visualise them. Using concrete materials or pictorial representations is vital for the children to really develop their number sense and their appreciation of these numbers and how they relate to,  and interact with, each other.

Number Theory in the real world

This topic is very interesting in the way it relates to spatial arrangements in real life. Triangular numbers can be seen in arrangements of bowling skittles, snooker balls, displays in the supermarket. Even numbers are evident wherever there is a pair.

Encourage the children to suggest examples and applications of odd and even numbers, prime and composite numbers, squared numbers, cubed numbers, etc. in the world around them. The idea of videos going viral on the internet, rumours or secrets being spread can all be related back to exponential numbers.

And as the children explore more, they may also begin to discover other fascinating connections between various types of numbers such as the total of two consecutive triangular numbers is always a square number and that double any triangular number is always a rectangular number. Again the emphasis should be on the children discovering this for themselves through engagement in rich tasks, such as the one shown below from the Early Finishers photocopiable in the TRB.

Points to note for the teacher

  • Counting/Natural numbers: When we talk about numbers in this topic we are referring to the set of counting numbers (i.e. 1, 2, 3, 4 … ) or natural numbers. These are whole numbers (no fractions or decimals) and do not include negative numbers. Since there is no universal agreement about whether to include zero in the set of natural numbers (some define the natural numbers to be the positive integers {1, 2, 3, …}, while for others the term designates the non-negative integers {0, 1, 2, 3, …}) in Operation Maths we only consider the positive integers, i.e. not including zero.
  • That which constitutes a rectangular number is undefined in the mathematics curriculum and the definition varies from source to source, with many conflicting with each other, particularly online. In Operation Maths, we have gone with the definition, as shown in the image below, given in the NCCA Bridging Glossary. This defines rectangular numbers as n x (n+1) which can also be expressed as n² + n; thus, the 6th rectangular number is 6 x 7 or 6² + 6 both of which equal 42.

  • Only whole numbers can be even or odd. The children may think that 1.2 or 4.36 are even numbers since their last digits is one of 0, 2, 4, 6, 8. However, only whole numbers can be denoted even or odd; you cannot create either an even or odd cube pattern (as shown in the Operation Maths 5 Pupils’ Book, p. 153), using 1 and a bit cubes.
  • Factors V multiplicands. In this sentence, 1.2 × 1.5 = 1.8, 1.2 and 1.5 are strictly speaking not factors even though they are being multiplied to produce a product. This is because factors are whole numbers, and when we ask for the factors of a particular number we want the whole number divisors of that number; if we were to include fractions (or decimal fractions) then the list would be impossibly long, complex and almost infinite. In the number sentence above, 1.2 and 1.5 are multiplicands or can also be referred to as decimal factors.
  • Factor pairs should be listed in a systematic way so as to identify all the pairs, i.e. not omit any, or equally not repeat pairs. Many children forget to include 1 and its pair as a factor, or any factors that are outside the traditional limits of the 10 × 10 factors,
    e.g. omitting 1 × 36, or 2 × 18 as factors of 36. The may also rewrite a pair, e.g. (2,3) and (3,2), not appreciating that because of the commutative property, order doesn’t matter and this is the same pair. To avoid these mistakes, demonstrate to the children how to identify factor pairs in a systematic way using a T-chart, as shown below.

Further Reading and Resources

  • Dear Family, your Operation Maths Guide to Number Theory includes practical suggestions for supporting children, and links to a huge suite of digital resources.
  • Virtual Maths Manipulatives for Algebra: Lots of tools that can be used in many different ways to explore odd and even numbers, factors, multiples, prime and composite numbers etc.
  • 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)
  • 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.
  • Explore exponential numbers via these lessons on the spreading of rumours and secrets.
  • Using Lego to develop math concepts: Read this article to discover ways to use Lego to explore square numbers, arrays and factors
  • NRICH: selection of problems, articles and games for prime numbers.
  • Check out this Number Theory Board on Pinterest

Digging Deeper into … Multiplication and Division (3rd – 6th classes)

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

Children are formally introduced to the operations of multiplication and division in third class, although their initial exploration of these concepts would have begun in 1st and 2nd classes with basic skip counting in jumps of 2s, 10s, 5s and, to a lesser extent, 3s and 4s. Thus, most children become quite comfortable with the number facts of twos, fives and tens quite quickly because these facts are more familiar to them. That said, when starting multiplication and division, and particularly when doing these groups of “easier” facts, it is still very important that the children are given ample time and experiences to develop a solid understanding of the concepts of multiplication and division themselves. This is why this topic is typically taught as a double (i.e. two week) chapter in Operation Maths.

Furthermore, as with the Operation Maths approach to addition and subtraction, multiplication and division is, for the most part, taught together, so as to reinforce them as related concepts that are also the inverse of each other. Thus, the initial activities in the Discovery Book, often require the children to reflect on their understanding of the concepts and to compare and contrast them.

As the children move through the classes, it is anticipated that they would begin to use this understanding of multiplication and division as the inverse of each other, to recognise the quotient (answer) in division as a missing factor in the inverse multiplication fact, e.g. asking what is 45 ÷ 9 is the same as asking how many groups of 9 equals 45 or [ ] × 9 = 45. This knowledge will also help as the children move towards more efficient recall of the basic division facts.

CPA Approach

As always, Operation Maths advocates a concrete–pictorial–abstract (CPA) approach to this topic. In 3rd class, this means the children will be moving from experiences with familiar objects that are already pregrouped (eg three wheels on a tricycle, four wheels on a car, ten toes on a person) and/or groups they create themselves using objects, to pictorial activities (e.g. where the children draw representations of the numbers using  pictures of the concrete materials) and finally to abstract exercises, where the focus is primarily on number sentences and the use of the multiplication and division symbols.

Using base ten blocks to demonstrate multiplication as an area array

Similarly, in 4th to 6th classes, the approach taken should start with concrete materials (eg base-ten blocks to represent multiplication using the area model, as shown above) as demonstrated by the images in the Pupil’s books and explained in the relevant Teacher’s Resource Book (TRB). As always, the emphasis is on the children understanding the process before being introduced to the formal algorithms (abstract stage). Otherwise, they are just being taught to use a procedure to produce an answer without really understanding why or how it works.

 

Moving on from the actual blocks; drawing an area model as a pictorial representation of the calculation.

Teachers should use opportunities such as questioning, conferencing etc. to assess their children’s understanding as opposed to just checking for a correct answer.  In particular, check does the child truly understand the purpose of the zero in the second line of the long multiplication algorithm; often when questioned a child will say they need a zero or the answer will be wrong, without appreciating that they are now multiplying by a multiple of ten, so the digits must be moved up a place, hence the necessary zero as placeholder. Therefore, in the example below, 14 is first multiplied by 7, giving a product of 98 on the first line, and then 14 is multiplied by 10 (not 1) giving a product of 140 on the second line. These partial products are then added to give a final total or final product. If in doubt about a child’s understanding, return to the concrete materials and/or pictorial phase again.  Based again on the area model of multiplication, the Partial Product finder tool from the Math Learning Centre can also be very useful to demonstrate this (shown below).

 

 

If you want even more examples of the effectiveness of using the area model of multiplication, please check out this video from James Tanton. The entire facebook post is accessible here

Similarly, when considering division, while the goal is for the children to be able to divide by two-digit numbers confidently, that doesn’t necessarily have to be via the formal long division algorithm. The long division algorithm has long been a controversial element in primary mathematics. Many feel it is counter-productive to be spending so much time in 5th and 6th class learning and teaching a procedural approach to solve written calculations, when the children will have continual access to a calculator for such calculations once they hit secondary school. In fact, in many other educational systems (UK and various US states included) they have abandoned the teaching of the long division algorithm completely in preference to the ‘chunking method’ (also known as the partial quotients method). However, if the method of the long division algorithm is taught using a CPA approach, as demonstrated in the Operation Maths Pupil’s book for 5th class, it can serve to enhance children’s understanding of number and operations in general.

 

Language & Properties

 

Consistent and accurate use of language is also an important way to emphasise the necessity of this interconnected CPA approach. The way that the number sentences for these operations are verbalised can unwittingly confuse children. If for  2 x 4, we say ‘two multiplied by four’, that implies 2 + 2 + 2 + 2, e.g. four plates with two cookies on each. However, if we say ‘two times/groups/rows of four’, that implies 4 + 4, e.g. two plates with four cookies on each. While it is understood that the order for multiplication doesn’t matter (i.e. the commutative property of multiplication) and that the answer for both of these is the same, children don’t inherently understand this, especially at the introductory stage. And to further confuse them, when represented concretely or pictorially, both images would also look different.

Therefore, for consistency, initially teachers and students  should use  the language of ‘groups/rows of’ to describe multiplication sentences and use the word times when verbalising the multiplication symbol (×). Similarly, in division we should initially use ‘shared/divided between’ and when using the division symbol (÷) we should say ‘divided into (… equal groups)’. When comfortable with the concept of division as sharing (i.e. 12 ÷ 2 = 6, 12 divided into 2 equal groups is 6 in each group) then the children can be introduced to the concept of division as repeated subtraction (12 ÷ 2 = 6, how many groups of 2 in 12 or how many times can I take  a group of 2 away from 12).

Once the children understand that they can use the commutative property to turn around facts in order to calculate more efficiently, they should be encouraged to recognise how multiplication sentences can be distributed into smaller groups, and that smaller groups can be combined to give the same total i.e  the distributive property of multiplication. It is hoped that they will begin to recognise where these properties can be applied in order to make calculations easier. Encourage the children to read distributed number sentences using the language of ‘groups of’ or ‘rows of’, as this can aid clarity, where the numbers and symbols all together can cause confusion. The 100 dots grids on the inside back covers of Operation Maths 3 and 4 can be extremely useful for the pupils to model various arrangements/arrays, while the teacher can use the 100 square eManipulative to replicate the children’s arrangements on the IWB.

Ultimately, being able to understand and apply the properties of multiplication greatly aids fact fluency. Understanding the distributive property, in particular, is key to understanding how the long multiplication algorithm works, i.e. that we multiply by the units and then multiply by the tens and then recombine these partial products. The distributive property and the ability to combine or break apart groups within factors also provides the children with strategies to mentally calculate products and quotients without having to use the standard written algorithms, for example using branching and T-charts, as featured in Operation Maths 5.

Thinking Strategies for the Basic Number Facts

When most of us were in school, multiplication and division was largely all about tables and learning off tables. Yet, research into this area in recent years suggests strongly that memorisation and rote learning is not an effective strategy; that even when successful (which may not happen for all), the children are learning off the basic number facts, without fully understanding them, and the connections both within and between these groups of facts. Therefore, while instant recall of the basic facts (up to 10 times and divided by 10) is a primary goal of both the mathematics curriculum and of Operation Maths, Operation Maths places significant emphasis on equipping the children with thinking strategies that will enable them recall the basic facts and to compute mentally, even with numbers that are outside the traditional limits of the tables.

Some of these strategies include utilising a child’s understanding of division as the inverse of multiplication, the properties of multiplication (e.g. identity, zero, commutative and distributive) which in themselves lead into exploring connections such as doubling and halving, thirding and trebling and facts as being one set more or less another fact etc. For example, consider 9 x 5

● 9 × 5 = 5 × 9 = half 90 =45 (communtative property & 5 times as half 10 times)
● 9 × 5 = (10 x 5)- (1 x 5) = 50 – 5 = 45 (9 times as one set less than 10 times)
● 9 × 5 = treble (3 × 5) = 3 × 15 = 45 (9 times as treble, treble)
● 9 × 5 = (3 × 5) + (6 × 5) = 15 + 30 = 45 (9 times as treble plus double the treble)

That said, it would be important that these strategies are not taught purely as tricks to help arrive at an answer. Rather, the emphasis should be on the children having sufficient concrete and pictorial experiences so that they can explore and deduce the connections between groups of facts for themselves and, thus, use what they know to solve what they didn’t know.

HINT: To find out more about Thinking Strategies for Multiplication and Division, please read on here: https://operationmaths.ie/thinking-strategies-for-multiplication-and-division-number-facts/

Mental strategies are as important as written methods

Similar to addition and subtraction, the aim is that the children will also become proficient calculating mentally, using such strategies as those outlined above, as well as using standard written methods. Once the child can use the thinking strategies to aid the recall of the basic number facts (previously referred to as tables), they can then progress to use these strategies to numbers outside of the traditional limits eg 5 x 18, 9 x 14 etc.

Again this would be very similar to the Number Talks approach to mental strategies for multiplication and division, and using the Number Talks resources accessible via the link above, alongside Operation Maths would greatly support the development of mental computation in any class. 

Another important mental strategy is the ability to estimate products and quotients. Similar to the approach to Addition and subtraction, Operation Maths places particular emphasis on the development of estimation skills for multiplication and division and introduces and develops specific estimation strategies as the books progress. Once again, the emphasis should be firstly on producing quick estimates based on rounding or finding compatible numbers and secondly, on examining quickly the reasonableness of the answers e.g. number of digits. Children sometimes feel that the teacher is adding extra work by asking their students to estimate; it is important they realise that, if done quickly, estimating can greatly improve their ability to recognise errors in their own or in other’s work. However, if a child continues to have difficulties with rounding or compatible numbers, e.g. if it is taking them too much time to create an estimate, then these children should be encouraged to use front-end estimations instead, only looking at the first digit, e.g. 1,643 × 6: think 1 thousand × 6 is 6,000. While these
estimates will be less accurate than those generated by rounding, they will give the children a sense of what answer to expect in a more time-efficient way. To find out more about some of the estimation strategies, read this post.

Problem-solving strategies

Once the children are familiar with the specific concepts of multiplication and division being taught, they should have opportunities to apply this understanding to a variety of problem solving situations, including word problems. These type of activities are typically part of the blue Work it out sections in the Pupils Books. For the children to develop their abilities to visualise the problems, they should be encouraged to use a variety of visual strategies, including bar models, branching, empty number lines, colour-coding specific operation phrases, and/or using guess and test. These various strategies are examined and developed at specific places throughout the Operation Maths books to allow the children to explore when and how they might best use them, but ultimately the child should be encouraged to use the strategy that works best for them.

In the multiplications and division chapters which are placed earlier in the school year, where the content is essentially consolidation of that covered in a previous class, some children may find the word problems quite easy to solve using a mental calculation and may be reluctant to ‘show their thinking’ in a visual/pictorial way, as they may feel that it slows them down. However, it is worth emphasising that they should get into good habits now, when the numbers involved are less challenging.

Further Reading and Resources:

  • Dear Family, your Operation Maths Guide to Multiplication & Division 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)
  • Mental Maths handbook for Multiplication and Division from the PDST
  • Number Talks book by Sherry Parrish
  • 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.
  • Multiplication and Division Board on Pinterest
  • Does the decimal point move? A one minute video which shows multiplying/dividing by 10/100 etc using the moving digits approach
  • Can I divide by zero? A one minute video which illustrates the difficulties that arise when we consider dividing by zero
  • Two more short videos from Graham Fletcher showing the progression of multiplication and division.

 

 

Digging Deeper into … Time (all classes)

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

Time is an integral element of our daily lives and therefore an essential, “need-to-know” topic in primary maths, particularly for those children with special educational needs or learning difficulties. However, it is also a very difficult mathematical concept, and one with which many children struggle, for a number of reasons:

  •  Time is abstract; it cannot be touched, or manipulated, or seen (although we can see its effects when we look in the mirror!).
  • The standard units of time do not reflect the familiar structures of our base-ten place value system i.e. 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days  = 1 week etc.
  • It can be displayed in analogue and digital forms and, furthermore, digital times can use a 12 or 24 hour system.
  • It is not uniform around the globe; each country belongs to a time zone and the time is different in each time zone.

Furthermore, there is a distinct difference between identifying a particular instant/moment in time (i.e. the skill of reading or telling the time) and understanding the concept of duration and the passage of time. In many cases, children may know how to tell the time, without having any real understanding of the concept of duration. Remember, that just because a child can read the digits on a digital clock/watch, or even read time from an analogue clock/watch, this does not mean they understand time as a concept.

As explained in all of the previous Digging Deeper posts, Operation Maths is based on a CPA approach. And, while we acknowledge that given the abstract nature of time is can be difficult to represent it concretely, Operation Maths does introduce the empty number line as a way to pictorially represent elapsed time (see below in post).

Telling Time

Telling time requires the children to:

  • develop an understanding of the size of the standard units of time eg days, weeks, hours, minutes
  • be able to estimate and measure using units of time
  • read and tell the time using both analogue and digital displays (12 hr & 24 hr).

The analogue clock has its own features that can further complicate matters; there is a “past” half and a “to” half; when reading times on the hour, the hour is said first (eg 3 o’clock) but when reading all other times the hour is said last (eg half past 3). Other valid questions that a child might have about the conventions of reading time include:

  • Why do we only say half past; why not half to? (Interesting point: the German for half past three, translates literally into English as half to four)
  • Why do we say half past; why not 30 past?
  • Why do we say quarter past/to; why not 15 past/to?

More often than not, when teaching analogue time, teachers tend to get the children to focus on the position of the long/minute hand and to use that to tell the time eg “if the minute hand is at 12, it is o’clock”, “if the minute hand points at 6, it is half past” etc. However, this type of explanation can in itself be very confusing, with many children interpreting half past any time as 6 o’clock, quarter past any time as 3 o’clock etc.

In fact, the first thing a child should be able to read is the bigger unit of time i.e. to identify the hour (hence, the first mention of telling time in our Primary Mathematics Curriculum is telling time to the hour, in senior infants . To do this, the children should, logically, look at the hour hand, which, although it is shorter than the minute hand, can often be wider on real analogue devices, emphasizing its significance. When the hour hand points straight at a number, then it is that time; when it points half way between two numbers, it is half past the previous hour (also the lesser number). In this way, the children will be enabled to tell time in relation to the hour eg “it’s nearly 2”, “it’s just gone past 7”, “it’s around half 5”, etc. Consider also how many children, when drawing hands on a clock to show time, will often have the hour hand pointing straight at the number even if it is half past, quarter past or a quarter to the hour. Focusing initially on the hour hand rather than the minute hand, highlights the fact that the hour hand also travels around the clock as the hour passes, and doesn’t jump from one hour to the next. A real or made clock with only the hour hand can be very useful here to teach this concept. Or use the Operation Maths Clock eManipulative (pictured below), and ask the children to look only at the red hour hand. The Two Clocks problem from NRICH can also be used to reinforce the importance of the hour hand.

Next, draw the children’s attention to the blue minute hand and to the blue minute markings around the edge of the clock. Logically, the minute hand has to be the longer hand because it points, beyond the numbers, at the minute markings which are furthest out from the centre, whereas the hour hand points to the hours (numbers) which are typically closer to the centre, and thus the hour hand is shorter. Emphasise that the minute hand enables us to become more accurate in our measurement of time. “O’clock” is actually an abbreviation of “of the clock”, so then when the minute hand points to the top of the clock , it is o’clock. Avoid saying “when the minute hand points to 12 it is o’clock” as the minute hand is actually pointing to the minute markings around the edge and not to the hours, which instead is the purpose of the hour hand.

At this point, it can be useful to use a cut-out paper circle (see below) to represent the clock and fold it in the centre to show both halves of the clock. Thus we can explain that when the minute hand points at the bottom/base of the clock, it is half past, as the minute hand has now passed through half of the clock. In a similar way, use the paper circle to make quarters and emphasise that when the minute hand has passed a quarter way through the hour, it is a quarter past, and when the minute hand is a quarter away from the next hour, it is quarter to. However, it should be acknowledged that, while it would be mathematically correct to say it is three quarters past the last hour (which prepares them for digital time), the convention used is to describe time in terms of how it relates to the next hour once it has passed the half-way point, at the bottom of the clock. Return to the paper circle at this point and label each half “past” and “to”. “Past” and “to” are also clearly labelled on the Operation Maths Clock eManipulative (see above), along with arrows to indicate the clockwise direction in which the hands travel.

When the children are ready, they should begin to measure time in minutes also (reading time in 5 minute intervals is on the curriculum for third class). Again, focus their attention on the blue minute hand and on the blue minute markings around the edge. In this way, the children may initially begin counting  the minutes in ones around the edge before realising that it is more efficient to count in groups of five, which co-incidentally, are also marked by the hour numbers. Creating and using peek-a-boo clocks can help reinforce this idea.

It is essential that when teaching digital time that the connection between analogue and digital is emphasised from the start. The Operation Maths Clock eManipulative is very useful in this regard as both clock types can be shown concurrently (see below). It is also includes the feature to hide/reveal the time in word form, as well as the feature to produce a random time on either clock, which can then be displayed manually on the other clock to match. For ease of use the teacher can also select the options that best suits the class level and ability eg hours only, time to half hours, quarter hours, five minutes, individual minutes.

Time as Duration

It is very important that the concept of time as duration is emphasised from the start. Duration of an event requires noting the starting and finishing points of time.
Developing a solid understanding of duration develops from a child’s experience  and understanding of:

  • Sequencing activities (eg pictures  of familiar/daily events, seasons etc)
  • The language of time such as before, after, soon, now, earlier, later, bedtime and lunchtime
  • The standard cycles of time (seconds, minutes, hours, days, months, seasons etc ) which in turn follows from the sequencing of daily events.
  • Measuring duration (using non-standard units initially, and then standard units)
    • How long does it take to …..? Estimate & measure
    • How many ….can you do in …? Estimate & measure
    • What if you do it faster/quicker…? (this in turn develops an understanding of the relationship between speed and time)
  • Comparing the duration of two events, (using non-standard units initially, and then standard units) eg what takes longer/shorter? How much longer/shorter is ….. than …..?

Developing an understanding of duration also requires the ability to visualise the passage of time in some way. For this purpose, empty number lines, one of the key problem solving strategies used in Operation Maths, are extremely useful. This can start with drawing an empty number line on the IWB or on the Operation Maths MWBs, to which class/question appropriate details can then be added:

  • What hour is after 9 o’clock? 11 o’clock? 12 o’clock?
  • Ann’s school starts at 9 o’clock. What time is it 2 hours later? 2 hours earlier? 5 hours later?
  • If Ann’s school finishes at 2:30 for how long is she at school?
  • Umair’s school starts at 9:30 and finishes at 3:00. For how long is he at school?
  • How long is it from the start of the 11 o’clock break to the start of the next break?

In Operation Maths, empty number lines are presented as a viable alternative, to the traditional column method approach, for calculating time. In many other countries, the traditional column method used for calculations involving units, ten, hundreds etc., is not encouraged, or not used at all, for calculations involving hours, minutes etc. However, as our Primary Mathematics Curriculum here in Ireland, still specifies the use of subtraction to solve elapsed time problems, Operations Maths has opted to present both ways in the books.

To view an excellent video of students solving elapsed time problems using an empty time line, please click here.

Other tips and suggestions for teaching time

  • Refer to, and use, aspects of time as much as possible during the school day, as appropriate to the class level e.g.:
    • Assign times for tasks and show interactive count-down timers on the IWB. This loop timer is particularly useful for timing stations in class.
    • Reference calendar facts such as the current, previous and next day, date, month, season, etc every morning.
    • Have a calendar visible in the classroom, marked with significant dates eg school play, outing, pupil birthdays etc. Ask the children to tell you how long it will be (in hours, days or weeks) until certain events (try to only use calendars that start with Monday as the first day)
  • Encourage the children to wear watches themselves, with a preference for analogue, as an awareness of analogue time better develops the children ability to appreciate the passage of time.
  • If buying a new clock for your classroom, try to ensure that it has the necessary features to help the children better understand how to read time.
  • When writing time in digital format always use a colon (as seen on digital displays) as opposed to a dot i.e. 10:30 as opposed to 10.30. Using a dot, which is identical to a decimal point, doesn’t help the child to recognise that the system of measuring time is inherently different from our base-ten place value system.

Further Reading and Resources:

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

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
  • Adding up in stages/sequencing (adding up in chunks)
  • Subtraction as take-away (removal/deducation)
  • Subtraction as difference (adding up/complementary addition)
  • 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.

 

Further Reading and Resources:

  • 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 … 2D Shapes (3rd – 6th)

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For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 2-D shapes, please check out the following post: Dear Family, your Operation Maths Guide to 2-D Shapes

Overview of 2D-Shapes:

The following are the new 2D shapes, to which the children are formally introduced, in the senior end of primary school:

  • 3rd class: hexagon
  • 4th class:  Equilateral, isosceles and scalene triangles; rhombus & parallelogram; pentagon and octagon
  • 5th class: Polygons, quadrilaterals, trapezium
  • 6th class: Kite*

* while the kite is not specified on the Primary Mathematics Curriculum (1999) for sixth class, it has been included in Operation Maths 6 as it features on the curricula for 5th/6th grade in many other countries.

As with every topic in Operation Maths, a CPA approach is also recommended for 2D shapes:

Concrete: Using concrete geometric shapes for classifying and sorting; identifying examples of 2D shapes and tessellations in the environment.
Pictorial: Tracing around shape templates to make reproductions that can be manipulated, folded, partitioned and combined; using lollipop sticks, geostrips or geoboards to create representations of 2D shapes; using the Operation Maths Scratch lessons accessible on edcolearning.ie to draw various shapes.
Abstract: Answering shape questions with no visual references/supports; suggesting the number of lines of symmetry on a shape without folding or drawing to explore the same; identifying the resulting shape when a given shape is rotated, flipped etc.

 

Properties of 2D Shapes:

Don’t let the list above, of 2D shapes by class, fool you; it shouldn’t create an incorrect impression that the primary focus is on identifying shapes, or that we should look at one type of 2D shape exclusive of others. Rather, the focus should be on the children examining the properties of each 2-D shape, describing it according to these properties and contrasting it with, and comparing it to, the properties of other shapes, rather than on just naming the shape. For example, what makes a square a square? How is a square similar to, or different from, a rectangle? Could an argument be made to say a square is also a rectangle? Could an argument be made to say a rectangle is also a square?

Therefore, any new 2D shapes that the children encounter should be compared to the 2D shapes with which they are already familiar. And, as the 2D shapes chapter in Operation Maths always follows on from the topic of Lines and Angles, when exploring properties, reference should also be made to the number and type of angles within the shape, the number and types of sides (parallel, perpendicular etc) and whether the shape is regular or irregular.

Common misconceptions:

Categorising 2D shapes  separately: As mentioned previously, children often don’t recognise a square as a type of rectangle, a rectangle as a type of parallelogram, a rhombus as a type of parallelogram etc. This can often be the case if the children are focused primarily on naming the shape and then compartmentalising it in a category, as opposed to examining its properties and exploring how it may have proprieties in common with other shapes.

It can be useful here to display 2D shapes to the class using a subgroup structure (like this one here) so that the children can appreciate how, for example, a square is also a rectangle, is also a parallelogram, is also a quadrilateral, is also a polygon.

Constancy of shapes: Many children don’t recognise that a  shape remains constant, irrespective of its placement in space. In particular, a square sitting on its vertex is often incorrectly labelled as a diamond. The children should be encouraged to draw or trace around shapes on their MWBs and then rotate the shape in order to appreciate its constancy.

Regularity: Children may not recognise a five-sided figure with sides of same length as being a regular shape. It is as if the terminology “regular” implies to them that the shape should be common-place i.e. regularly occurring. For this same reason, a child will often say a rectangle and a circle are regular shapes, given their familiarity with these shapes from the junior classes, even though they are officially classified as irregular shapes. Challenging this misconception will require plenty of sorting activities where shapes are classified as regular or irregular (see Ready to Go activities below).

When creating a class display of shapes eg rectangles, triangles, etc., instead of using just one qualifying shape to illustrate the term, use many and use varied ones. Enlist the help of the class: “I want to make a display of triangles/parallelograms but I want the triangles/parallelograms to all be different. Can you draw and cut some out for me?” Such an activity would quickly reveal those who appreciate the required properties for each shape and those who don’t. Remember to also position the shapes in various ways so as to reinforce that the shape remains constant, irrespective of placement.

Identifying 3D objects as 2D shapes: This is a very problematic area. It often happens that when asked to find a circle in the environment, a child suggests a ball (a sphere) or a cube might be suggested as a square. When asking to identify 2D shapes in the classroom or at home, we must be careful how we respond to the answers so as not to reinforce these misconceptions. For example, if a child suggests that the door is a rectangle, when it is in fact a cuboid, emphasise that a part of the door is rectangular eg

  • Which part of the door is like a rectangle?
  • Are there any other parts of the door that are like a rectangle?
  • Can you see any other rectangles on the outside of the door? How many?
  • Are they all the same or different?

Asking the children to draw around solid 3D objects in order to produce flat 2D shapes can also be useful here.

Coordinates (6th class)

The concept of plotting and reading coordinates is introduced in 6th Class. There are plenty of examples of coordinates in the children’s environment, e.g. map reading, car parks and board games such as chess and Battleship. Allow the children to practise reading coordinates on maps and on board games, then progress to using two digit coordinates in maths. Make sure they first read the horizontal coordinate and then the vertical coordinate.

Operation Maths Digital Resources:

Don’t forget to access the linked digital activities on the digital version of the Pupil’s book, available on edcolearning.ie . These include:

Ready to go Activities: based on the Sorting eManipulative, these enable the various shapes to be sorted according to class-appropriate criteria, or enable tessellating patterns to be made. The Ready to go activities all have suggested questions inbuilt on the left-hand side of the screen that the teacher can just click to reveal and hide. Remember, when sorting, the focus should be on the properties of the shapes not their names; that said, you can also ask the children to identify the shapes, if known, as an extra dimension to the activity.

Create activities: (all classes) again using the Sorting eManipulative, these are less structured that their Ready to go counterparts. Instead, the teacher should click on the yellow “Create new example” button on the bottom of the screen, and then use the sorting eManipulative to explore the shapes as they see fit. The teacher can use a previous Ready to go activity to inspire the create activity or come up with a completely different activity of their own using the almost limitless possibilities of the sorting eManipulative.

Write-Hide-Show videos: These explore tessellations (3rd class) and different types of triangles (5th class). They encourage the children to look and respond to the questions by answering orally or on their MWBs.

Maths Around Us video (6th class): which examines different types of triangles from the environment.

Scratch-based programming lessons with instructions on how to draw 2D shapes  (3rd class) and hexagons (3rd, 4th, 5th classes), scalene triangle, pentagon and octagon (4th class), different types of triangles (5th class) and plot 2D shapes on a grid (6th class).

 

Further Reading and Resources

Digging Deeper into … Representing and Interpreting Data (3rd-6th)

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Category : Uncategorized

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:

  • What is the most common eye/hair colour in our class?
  • Which fruit/pet do we prefer?
  • How did we come to school today?
  • What candidate would we vote for?
  • What is the temperature each day?
  • How many children are absent from school every day?

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. Thus, 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. And, in the case of the questions about temperature and number of absences, the children may begin to appreciate that it is too much to give the specific details for each individual day and that a figure to represent a larger set of numbers (eg the average) is preferable in some situations.

Content overview

A quick glance at the curriculum content for representing and interpreting data for these classes, reveals the following:
3rd class: pictograms, block graphs, bar charts
4th class: pictograms, block graphs, bar charts and bar-line graphs incorporating the scales 1:2, 1:5, 1:10, and 1:100
5th class: pictograms, single and multiple bar charts and simple pie charts; calculating averages
6th class: pie charts and trend graphs; calculating averages

In Operation Maths for 3rd and 4th classes, representing and interpreting data is specifically taught in September, at the beginning of the school year, so that the children are enabled to incorporate these skills into other subject areas where possible e.g. reading and interpreting tables and graphs, collecting and displaying data in science, geography etc.

In 5th and 6th classes, representing and interpreting data is taught later in the school year, after the children have encountered degrees in lines and angles and the circle in 2D shapes, as this content is necessary prerequisite knowledge for creating pie charts. In these classes, representing and interpreting data is also taught as a double chapter (two week block), to allow for the extra time required to explore averages.

CPA

As with every topic in Operation Maths, a CPA approach is also recommended for representing and interpreting data:

Concrete: Using real objects to sort and classify eg the number of different colour crayons in a box, the different type of PE equipment in the hall etc; using unifix cubes, blocks, cuisinere rods etc to represent data; using cubes to introduce and explore the calculation of averages.
Pictorial: using multiple copies of identical images to make pictograms; using identical cut out squares/rectangles to make block graphs etc, using folded circles to make pie charts, using bar models to calculate averages.
Abstract: the final stage, where the focus is primarily on numbers and/or digits eg reading and interpreting the scale on a graph where all the scale intervals are not given; calculating averages without pictorial or concrete supports.

Interpreting data

For children to become comfortable interpreting tables and graphs it is vital that they have plenty of opportunities to look at and read a variety of tables and graphs. This shouldn’t be limited to just the tables and graphs in their maths books. In particular, data sets that are relevant to them, such as soccer league tables can be a great way to encourage the children to appreciate how relevant this strand units is to them.

Utilize every opportunity to expose them to real-life examples of data from print and digital media and use purposeful questions to highlight the features of the graph:

  • What is the title of this graph/chart?
  • How is the information displayed? Horizontally or vertically?
  • What type of chart/graph was used?
  • Why do you think this graph type was chosen? What other types would have been suitable?
  • What key information is required to interpret the data (eg scale intervals, labels on the axes, a key for piecharts)?
  • Is there information missing that would have been useful to get a better insight into the data?

The children can be asked to create questions based on the graph/chart and swap with a partner to answer. When they become adept at producing charts themselves (see next section) they can also be asked to represent the data using a different chart type.

One of the most common mistakes that children make when interpreting graphs is misreading the scale. Always draw children’s attention to this first, and ask them to identify the scale interval and what it means for the bars/blocks/points etc on the graph. The graph quiz on That Quiz provides lots of extra practice for this skill. The quizzes are also very customisable, with options to show pictograms, bar charts, trend graphs (line) and pie charts (circle), easier or normal content, and a variety of question types. Another similar activity is this one from MathsFrame which offers three different levels of questions on bar charts.

A very interesting  and very different way to explore interpreting data is to show the children graphs where much of the key information is missing initially, but is then slowly revealed as the children share their thoughts and ideas. Following on from Brian Bushart’s work on numberless word problems, many teachers have used graphs to create “slow reveal” activities or “notice and wonder graphs”, and have very generously shared these online for other teachers to use. Some of these include:

 

Representing data

As mentioned previously, where suitable children should begin to represent data themselves using concrete materials. They can build block graphs using cubes or blocks, laid flat on a piece of paper or their Operation Maths MWBs. These should all start from the same baseline and the children should also write in labels for the axes and a title.

As a development, they can then trace around the stacks of cubes and remove the cubes to have a pictorial representation of the concrete. Using cubes like this to represent 1:1 quantities can in turn lead children to see a need for one cube to represent more than one, ie scales of 1:10, 1:5 etc, especially if there are not enough cubes to represent the data or there is not enough space.

The next step could be to have small squares or rectangles of identical pieces of paper which can then be pasted onto a page to display the information. This can work particularly well for pie charts; cut out a circle of paper and divide it by folding into eighths; the circle can be left whole and the folds outlined in pencil/marker or the eighths can be cut up. A groups of eight children can then use either of these to show data like their favourite ice-cream flavour or TV programme. In this case, because the amount of data gathered is limited, the choices/categories should be limited, also, to three or four.

      

If the children are also collecting the data to make a graph or chart, they will need to come up with a system to accurately collect and record this data. This will usually involve compiling a type of table with three columns; the first column to list the categories, the second to record tally marks and the third to total the tally marks. When introducing tally systems the children could use lollipop sticks to explore and make tally marks.

For children, drawing their own graphs can present many difficulties. Some common mistakes that can be made include:

  • Incorrectly transferring the data from the table to the graph.
  • Omitting the graph title and/or category titles on the axes.
  • Using an inappropriate scale for a specific graph.
  • Not setting the scale at regular, even intervals
  • Zero being incorrectly located somewhere other than at the base line/axis.

And in other cases, it can just be a lack of neatness and exactness that reduces the quality, and readability, of a hand-draw graph. To overcome the difficulties associated with hand-draw graphs, the children could use either an online or offline computer application, all of which can produce very impressive results. Listed below are a small sample of those available; click on any of the links to access a tutorial or the application itself.

 

Calculating averages

Averages are introduced for the first time in 5th class and the children should have ample opportunities to explore this concept concretely and pictorially, before being given the formula to calculate the average of a set of numbers. Initially, the concept should be introduced as sharing amounts out to be fair/balanced:

Through plenty of concrete and pictorial opportunities to balance these separate quantities, it is hoped that the children begin to see a connection between the total number of items and the balanced quantity or average:

Bar models, one of the key problem-solving strategies used in Operation Maths, are very useful here, where comparison models can be used to compare the total of the averaged quantities with the total of the individual quantities. They are also used in Operation Maths 6 to calculate the extra number(s) when the average increases or decreases, a concept which can be very difficult to reason if no pictorial structures are used to help visualise the relationships.

You can also check out this video to see how bar models can be used to solve averages:

Further Reading and Resources

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