# Digging Deeper into … Chance (3rd – 6th)

## Digging Deeper into … Chance (3rd – 6th)

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

Chance is one of the most fascinating areas of primary mathematics, since it is concerned with the outcomes of random processes. Thus, the conceptual foundations for areas of mathematics such as probability and combinatorics, can be found in this strand unit.

### The big ideas about Chance:

• When considering random events and/or processes, we can use what we know (eg past experience and/or knowledge of the variables involved) to estimate/predict the likely outcome(s).
• If we identify all the possible outcomes in advance,  we can refine and/or express our prediction using mathematical language.
• However, no matter how accurate the mathematical prediction, the actual outcome(s) is not certain (except in the unlikely case where there is only one possible outcome); that is the element of chance!
• If we collate the results from repeated identical investigations of a specific random process, the actual outcomes (experimental probability) are more likely to reflect the original mathematical predictions (theoretical probability).

### Predicting Outcomes: Terminology

When beginning to discuss and predict the likelihood of various outcomes,  the initial focus should be on the language of chance, and the terminology that accompanies it.

It can be very useful for the children to identify the various terminology, to discuss their interpretation of it and to explore the contexts in which the terminology is used in everyday parlance.

And while some of the phrases are more objective (e.g. impossible, never, certain, sure, definite), much of the language can be more ambiguous and is open to personal interpretation (possible, might, there’s a chance, (highly) likely, (highly) unlikely, not sure, uncertain).

FACT: To avoid ambiguity, some organisations have agreed on a consensus that equates this terminology with a fractional expression or percentage; you can view one such consensus here.

It can be helpful to try to organise this language across a continuum for the children to interpret and establish their meanings in relation to the other phrases. Ask the children to identify terminology that is used when describing the likelihood of something occurring. Use questions/statements to elicit from the children the vocabulary for chance that they already have; this can be the language that they would use to answer the questions from the text above or could be from their responses to questions such as the following:

• What is the chance that it will rain today?
• What is the chance that it will be hot today?
• What is the chance that it will be dark tonight?
• What chance does my team have of winning the league?
• What chance does my county have of winning the All-Ireland Championship?

Ask the children to write this terminology on pieces/slips of paper. Sort the pieces of paper into groups and/or order them along a line (continuum), as shown in the images below, with words that have similar or identical meaning together.

This task is a perfect example of a low threshold, high ceiling task, in that all children can participate and there is no limit to the complexity of terminology that can be incorporated. If mathematical values such as percentages and/or fractions (eg 1 in 2 chance) are suggested, the children should be encouraged to incorporate these, as they see fit.

Indeed, in fifth and sixth class the children should be encouraged to use a continuum which is graded from 0-100% and/or 0-1, and to associate and align the vocabulary with mathematical values (eg impossible/never =0%, might or might not/even chance = 50%, definite/certain = 100% etc).

### Predicting Outcomes Mathematically

Irrespective of whether it is tossing a coin, rolling a dice, spinning a spinner, picking from a bag, choosing a card, etc., the children should always be encouraged to identify all the possible outcomes, to predict outcomes that are more or less likely, and to justify their predictions.

The children can also be encouraged to make more mathematical predictions based on their understanding of the variables involved e.g. if we repeated this investigation 30 times, how many times would you expect each colour would be picked? What about 60 times? 120 times? Express the fraction of the total number number of “picks”, that you would expect for each colour. Can you express any of these as a percentage?

When predicting the outcomes of random processes that involve a combination of variables, it can be very useful to use a type of pictorial structure, such as branching (NB these can also be referred to as tree diagrams), to illustrate the possible outcomes. For example, when predicting the outcomes of a double coin toss, children will often think that each of the three outcomes have an equal chance, when in fact there is double the chance (ie 2 in 4 or 1 in 2 chance) of getting a heads and tails combination, than either both heads or both tails (see diagram below).

However, it is worth noting that, unless the children come up with a similar structure to predict outcomes of combinations, it is preferable to hold back on showing such a structure until they have conducted an investigation, similar to above, where their predicted outcomes did not align to the actual outcomes.

### Conducting the investigations

Once all appropriate predictions have been recorded, we can move on to the most exciting part, the investigating! When conducting chance investigations, it is important that the children recognise that that they need to be conducted fairly and recorded clearly, similar to scientific investigations.

Encourage the children to consider what factors need to be kept the same each time, and how practices could affect the reliability of the results eg:

• When picking items (eg cubes from a bag, cards from a deck) does the chosen item need to be returned each time? Why/why not?
• How many times does an investigation needs to be repeated in order to get a reliable result?

To generate sufficient data, while not spending too much time on each investigation, ten can be a suitable number of turns per child. It can also be a good idea to organise the children into groups of three with rotating roles eg the first child has their turn, the second child records the outcome of each turn and the third child keeps count of how many turns the first child has had, and roles are rotated after ten turns.

### Recording and reflecting on results

As mentioned previously, the children should be encouraged to consider how best to record results. Tally charts and frequency tables can be very useful and link in well with the strand unit of Representing and Interpreting Data. Results of investigations can be displayed in various types of graphs and charts. Children in fifth and sixth classes could also be asked to calculate the average value for each outcome, when all the results of a class group are considered; for example, in the double coin toss, what was the average number of heads, tails and heads-tails combination per group.

Once the results have been collated, it is very important that the children be given time to reflect on the results and to compare them to their predictions. While we would expect an equal number of heads and tails in a single coin toss (ie theoretical probability), the actual results may not resemble these predictions (experimental probability). Such is the element of chance! And this can be a difficult concept for the children to accept, particularly the notion that, even though the mathematics behind their predictions was accurate, the actual outcomes are different.

To explore this further, using a spreadsheet, such as Google Sheets or Microsoft Excel, to collate the results of the entire class can be a great way demonstrate, that when we combine all the investigations, experimental probability (ie the results) is more likely to mirror theoretical probability (the predictions). This can often help reassure the children that the “maths” behind this does indeed work!

TIP: To make life easier for you, we have created a sample spreadsheet for the Double Coin Toss, please click on the link to view (and save/copy). For further information on the values of using spreadsheets to record results please check out this informative article on Probability Experiments with Shared Spreadsheets from NCTM.

• Dear Family, your Operation Maths Guide to Chance includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• The Operation Maths Digital Resources have specific resources designed to support this strand unit. Full details of these can be found in your TRB. Click here for the Quick Start Guide to the Digital Resources.
• Virtual Maths Manipulatives for Chance: Lots of tools that can be used in many different ways to explore this concept.
• PDST Data and Chance Handbook for Teachers
• Data and chance in the world around us: materials from the PDST workshop
• NRICH: selection of problems, articles and games for chance
• Playing dice, card, spinner games, or indeed any type of chance-based games, can be a great way to get students thinking about probability, while also providing practice with mental computation, estimation, subitising and experience of problem-solving via strategic thinking.
• Don’t forget to check out the games bank in your Operation Maths TRB and/or the last page of the Number Facts books for examples and ideas.
• Check out this Mathwire page for more games that focus on probability.
• For a fifth and sixth class who are exploring combinations, Mashup Math has two excellent videos (view both below) which demonstrate how tree diagrams and area models can be used to identify all possible combinations; both video use contexts to which the children could readily relate.

## Digging Deeper into … 2-D Shapes and 3-D Objects (infants to second class)

Category : Uncategorized

For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 2-D shapes and 3-D objects, please check out the following posts:

Dear Family, your Operation Maths Guide to 2-D Shapes

Dear Family, your Operation Maths Guide to 3-D Objects

### 3-D shapes or 3-D objects?

In the PDST Shape and Space manual, it is suggested that “using the word ‘shape’ to describe both 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 titled 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 1999 Primary Maths curriculum.

### So what first? 2-D or 3-D?

2-D and 3-D objects are very inter-related, to the point that there is often much debate about which of the topic should be taught first; 2-D shapes, 3-D objects or teach them both concurrently.

Since 2-D shapes are lacking the third dimension of depth or height that their 3-D relations possess, this makes them quite abstract as only flat, drawn/printed shapes are truly 2-D. Whereas, 3-D objects can be picked up, manipulated, used for constructions etc., making them much more suited to the concrete learning experiences that are essential in the early years. They are the objects that we find in the real-world. Thus, since 2-D shapes are only flat representations of the faces of 3-D objects, it could be argued that it would be more logical, and more in line with the concrete-pictorial-abstract (CPA) approach, to teach about 3-D objects before 2-D shapes.

On the other-hand, it could be argued that 2-D shapes should be taught first as it is likely than young children would be more familiar with them. For example, the vocabulary of 2-D shapes features more regularly in common speak than the vocabulary of 3-D objects. Many children will likely have encountered many 2-D shapes from picture books and patterns around their homes, etc. And so, it remains inconclusive as to which order of progression is most beneficial!

In the Operation Maths books, the children meet the specific topic of 2-D shapes prior to that of 3-D objects each school year. However, it is envisaged that by the time the children in the junior classes are formally engaging with 2-D shapes, they have already encountered and informally explored both 2-D shapes and 3-D objects via the monthly themes (laid out in the long-term and short-term plans of each TRB) and in the suggested Aistear play activities (detailed also in the TRB) of which, the Aistear theme of construction is particularly relevant.

### Infant classes

Whether considering 2-D shapes or 3-D objects, the suggested progression within each topic is very similar:

• Undirected play
• Sorting and ordering activities
• Building and making (including making patterns)
• Identifying

Undirected play may include sand and water play, use of formal construction toys, constructing using “junk” or found materials; any activities that allow the children to handle and manipulate shapes and objects.
In the Operation Maths TRBs for junior and senior infants there are ample suggestions for suitable activities, under the headings of various themes. “Undirected play” does not imply that the teacher is superfluous to the process; rather while the children are the instigators, the teacher can play a vital role, observing the way in which the children interact with the materials, and asking the children to explain what they did, how they did it and why they did it that way. This can be a great way to assess the prior knowledge and language that the children may already have.

Sorting and ordering activities include the Early Mathematical Activities (EMA) used early on in the infant classes; thus it is likely that shapes and/or objects have already been used as part of these activities, for example sorting and matching according to colour, size etc; ordering according to length/height etc.

At this point, the children should also be prompted to sort the shapes and objects according to their respective properties as relevant and appropriate:

• Sort 2-D shapes according to the number of corners and the number and type of sides (straight, curved or both; sides that are different or the same).
• Sort 3-D objects according to those that roll/do not roll, slide/do not slide, build/do not build, are hollow/solid; as a development, according to the number of corners and the number and type of faces and edges (please see end of post for more information on faces and edges).
• The teacher can also isolate shapes/objects to create sets and then ask the children to identify the rule of the set: “What’s my rule?” (see image above). The children can also be encouraged to play the “What’s my rule?” game in groups.
• Isolate a particular shape/object in the room and ask the children to locate others that are the same/similar and make a set of like objects/shapes.
• The children may also be naming the shapes as part of these explorations; however this is not necessary as it is more important that they appreciate the similarities and difference between shapes, rather than identifying them.

Building and making with shapes/objects may have already been explored informally as part of the undirected play phase. The purpose now is to develop this into more formal teacher-directed tasks and activities:

• Build the tallest building/castle that you can. What objects did you use/not use and why?
• Dip a face of a 3-D object in paint and use it to print. Make a pattern using the prints. What do you notice?
• Try printing with different faces of the same 3-D object. Are the resulting prints the same or different?
• Push 3-D objects into sand/plasticine to make imprints. Or (if able) trace around the 3-D objects on paper to make designs.
• Use cut-out shapes, gummed shapes, tangrams and/or pattern blocks to make pictures and patterns.
• Use the shapes to cover surface of your book/mini-white board; which shapes did you use/not use and why?
• Combine two shapes/objects to make a new shape/object.

As part of the building and making activities the children may begin to realise how certain shapes/objects can be combined to become other shapes/objects. Similarly, through the reverse of these activities, and other shape cutting activities, the children should begin to realise that shapes can also be separated (partitioned) to reveal new shapes. This can include deconstructing 3-D objects to reveal their net. These activities can be revisited once the children can also name the shapes/objects, so as to arrive at certain understandings and become more accurate with mathematical language eg that two squares can make a rectangle; that, when using tangrams, two of the same size triangles can be rearranged to make a square, a larger triangle etc.

Identifying the specific shapes/objects evolves from the previous activities as the children begin to realise that it is the specific properties and attributes of a shape/object that defines it, eg all shapes with three corners (and three sides) are triangles, irrespective of their size or colour and irrespective of the measure of sides and corners (later to be referred to as angles). Activities which will serve to reinforce this include “Guess the shape/object” using descriptions (see below), guessing unseen shapes/objects from touch (eg in a feely bag), locating a specific shape from a collection using touch alone.
Through the experiences of printing and imprinting with the 3-D objects, it is also hoped that the children realise that the flat faces of 3-D objects are in fact 2-D shapes.

### First and Second classes

The children in these classes will continue to sort, describe, compare and name shapes as done in infants, but to now also include new shapes and objects i.e. semi-circle (1st), oval and cone (2nd). They will continue to construct and make shapes, extending this to creating and drawing the shapes themselves.

They will further explore the combining and partitioning of 2-D shapes, and this understanding will extend to include the fractions of halves (1st) and quarters (2nd). The properties of 2-D shapes will be further explored
in second class via the stand units of symmetry, angles and area (i.e. tessellating 2-D shapes)

Q: 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 it may be argued 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, or printing, 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).

## Digging Deeper into … Comparing and Ordering (infants to second class)

Category : Uncategorized

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

Of all the strand units in maths, this topic is one that is very close to the hearts of almost all young children:

• “She’s got more than me! That’s not fair!”
• “I want to be first!”
• “I want to be the biggest!”

This strand unit evolves from the separate strand units of comparing and ordering that, along with the other two strand units of classifying and matching, make up the strand of early mathematical activities. The content objectives for this strand unit are quite similar across the four junior classes, with the main difference being the specific number limits for each class level:

Number > Comparing and Ordering >The child shall be enabled to:

• compare equivalent and non-equivalent sets (to include the symbols <, >, = in second class)
• order sets of objects by number (infants to first only)
• use the language of ordinal number

### Comparing

As mentioned above, even from when they are very young, most children are quite adept at comparing what he/she has with that of another.

As part of the strand early mathematical activities (i.e. pre-number) the children will already have had experience comparing sets by quantity (but without counting) i.e. identifying which of two sets has an obvious amount more (or less) than another. They will also have been identifying two sets/objects as being the same or different.

In Junior Infants, once children are comfortable establishing the cardinality of sets up to five, the next step is comparing and ordering sets of objects up to five. Since the amount of these sets may often only differ by just one or two, then it is not very obvious, from a visual point of view, which one has more or less. Comparing two similar sized sets requires that the child:

• Can identify (and, later, write) the correct numeral for that set
• Understands one-to-one correspondence, and using this can match the items in the two sets, so as to establish which one has more or less
• Understands the conservation of number i.e. that a short line of five objects situated close together still has more than a longer line of four objects further apart.
• Does not assume that the quantity of a set with objects bigger (or smaller) in size must be greater (or less) than the other set.

### How many more?

Once a child is able to identify the greater set, the next step is to be able to state the difference between the sets i.e. how many more plates than cups? This can be a very difficult concept, with which children can struggle for many years.

As with the entire Operation Maths programme, a CPA approach is recommended when teaching this concept and, in particular, to use that which is most familiar to the children:

• Use items that typically go together eg knives and forks, cups and saucers/plates, children and chairs/coats/school bags. Take a number of each and ask the children to suggest how we could ascertain the number of each. If not suggested by the children, the teacher should demonstrate how to set out the items in groups together eg the first knife with the first fork, the second knife with the second fork etc. If the quantities of each are not equal/the same, ask the children to explain how many more of the lesser quantity is required AND to explain how many extra items there are in the larger amount.
• In a mixed classroom, use girls and boys. Call up a random group of children, ask the boys to line up at the top of the room, and the girls to line up in separate line beside them, so that, where possible, each child is adjacent to one other child in the other line (if you are lucky enough to have square tiles on your floor, ensure that there is a child standing in each square space). Ask the children to identify the children who have a match/partner on the other line and the number of children who do not have a match/partner on the other line. This activity could also be repeated using dolls and teddies, toy farm or zoo animals, attribute bears etc.
• Use concrete manipulatives and pictures. Start with only two sets initially. Impress up on the children that the easiest way to see the comparison is to “line up” the objects, was done with the children previously. Use a grid of squares* to help with this. Once again, ask the children to identify where there is a “partner” fruit on the other line and the number of fruit that do not have a “partner” on the other line. These are the extras. How many more (extra) bananas than  apples? How many more (extra) bananas than  strawberries?  *The 5×5 grid on  the Operation Maths Sorting eManipulative is very useful here. The Operation Maths 100 Square eManipulative can also be used; select to show counters only and line up two (or more) rows or columns of different colours.
• Ultimately, it is hoped that the children realise that it is not necessary to establish the exact amount of each set to be able to establish the difference between each set. In the example above, there are two more bananas than strawberries, and it is not necessary to identify the number of each fruit to establish this. This encourages the children to develop efficiency and flexibility in their approaches.
• As the children move into first and second class, they should still be encouraged to “line up” the sets. If comparing the number of items in two static sets that cannot be lined up, eg an image in their books, the children can represent the number of items in each set using cubes and these cubes can then be lined up to make it easier to identify the difference between each set. This would link very well with their experiences of comparing quantities in pictograms and block graphs from the strand of Data.

It is important that teachers are aware that establishing the extra number in the larger/greater set and establishing how many less/fewer in the smaller/lesser set requires the children comparing the amounts in two different ways. In the example above, to identify how many more bananas there are than strawberries, requires identifying the number of bananas for which there are no corresponding strawberries. However, to identify how many fewer strawberries there are than bananas, requires identifying the number of empty spaces in the strawberries that there are, opposite the extra bananas. While the answer is the same both time, the route to the answer is different, and the latter approach requires the children to count empty spaces, which is more challenging due to its abstractness.

In second class, the children will begin to use the inequalities symbols (<, >). Many children will struggle with selecting the correct symbol to use, even if they can identify the larger or smaller quantity. Thus flashcards or reference cards such as the ones at this link can be very useful to connect both the language and the symbol. Interactive quizzes like this one from That Quiz or this one from ixl.ie can provide opportunities for extra practice. However, as emphasised previously, it may still be necessary to use a visual representation of both numbers being compared, for example using stacks of cubes, base ten blocks, straws or base ten money (10c and 1c coins). In this way, the children are now beginning to use their place value understanding also to compare quantities. As well as using the actual concrete materials, the Sorting eManipulative can be used to demonstrate how to do this using images of base ten materials; see Ready to go activities 2.3 and 2.4 as examples (screenshots below).

Hint: Developing the children’s ability to compare, will also be of benefit when they encounter the concept of subtraction as difference (as opposed to subtraction as deduction/take-away) and of further benefit when they are introduced to comparison bar models in third class up

### Ordering

As part of a early mathematical activities, the children will already have experienced ordering objects by length, size etc. Now, they are extending this understanding to order by quantity.

In Junior Infants, once the children are able to count individual sets of up to five objects, this enables them to start ordering the sets of objects.

Counting and numeration are both very important when it comes to ordering:

• The children are beginning to understand how higher numbers correlate with greater numbers of objects and vice versa.
• When ordering sets we must also consider the number word sequence i.e. number five comes after the number four so five must be a greater amount than four.

### Ordinal numbers

The nature of the English words for the ordinal numbers (first, second, third, fourth etc) and the nature of their abbreviated forms (1st, 2nd, 3rd, 4th etc) can pose significant difficulties for children as, at first glance, there appears to be little correspondence between the forms, and the abbreviations may not appear to follow any rule or pattern. Another difficulty lies in the apparent contradiction between ordinal numbers and cardinal numbers; it is typically better to have 10 rather than 1 of anything, but it is typically better to be 1st rather than 10th in any competitive activity.

• Initially the focus should be on the spoken words only and the activities used should reflect this eg lining up children at the classroom door, asking the rest of the class to identify who is first, who is second, third, last etc.
• When ready, flashcards of the ordinal words should be introduced and these can be incorporated into the familiar activities eg the flashcard with “first” can be given to a child who must give it to the child in that position in the line.
• It is better to avoid using the abbreviations until first class and it is also better to start with the words, fourth, sixth seventh and tenth. Write the word fourth on the board and establish that the children can read and understand the word. Explain that for speed we want to find a quicker way to write/indicate this position and ask them to suggest what might be written to replace the underlined part of the word (ie 4th). Repeat this with the ordinal words sixth, seventh and tenth. Ask the children to suggest how fifth, eighth and ninth might be abbreviated and then finally ask for suggestions for the words first, second and third; ultimately, tell them the correct answers if they do not arrive at them themselves. In this way, the children are being prompted to discover the system of abbreviations that we use, as opposed to being just told.

Hint: For first and second classes, there is a list of online interactive games here which will help as extra practice. There are also lots of useful videos on YouTube etc; just search for “ordinal numbers”.

## Digging Deeper into … Spatial Awareness

Category : Uncategorized

Spatial awareness…being able to describe the position of something/someone in relation to another using words and/or gestures, and being able to represent spaces and locations using models and/or drawings, may, at first glance, appear to have more in common with communication and geography, than with maths. However, the concepts of spatial awareness lay the foundations for all geometric thinking, be it at upper primary, secondary or an even higher level.

Essentially the children need to develop an understanding that:

• The spatial relationships between objects and places can be described and represented.
• These relationships may be viewed, described and represented differently depending on the perspective of the viewer (in particular, consider left and right).
• Developing the ability to mentally visualise the representations will enhance a person’s ability to picture how a shape will look when rotated when turned, flipped etc.

A synopsis of the curriculum objectives for infants to second class, state that the children should be enabled to:

• explore, discuss, develop and use the vocabulary of spatial relations (describing both position and direction/movement)
• explore closed shapes and open shapes and make body shapes
• give and follow simple directions (first and second class), including turning directions using half and quarter turns (second class only)
• explore and solve practical problems (first and second class)

In the case of the practical problems, this could include completing a jigsaw or a tangram puzzle, using mazes, grids, board games and or exploring basic coding eg via coding programs and apps, such as Lightbot, and more hand-on devices such as BeeBots.

### Moving through space

Since spatial awareness requires an understanding of using  space and moving through space, the majority of the activities should be active ones, where the children are moving around. This is where the suggested activities in the Operation Maths Teachers Resource Book (TRB) become extremely useful, such as the examples below.

Much of the language development in this strand unit can be reinforced via activities in PE (Orienteering) and Geography (mapping).

### Digital Resources

While activities incorporating physical movement are preferable, the Operation Maths digital resources on edcolearning, provide a worthwhile alternative and add variety. The Ready to Go activities below, as the phrase says are “exactly what they say on the tin”; the teacher need only click on the relevant icon in the digital version of the pupil’s book to open the activity, and the accompanying suggested questions are quickly viewable along the side menu. A full description of the activity, including the questions, is also given in the TRB.

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

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).

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.

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

Category : Uncategorized

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

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.

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.

• 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

## Digging Deeper into … Counting and Numeration

Category : Uncategorized

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.

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)

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:

• 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

## Digging Deeper into … Capacity (all classes)

Category : Uncategorized

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

## Digging Deeper into …. Weight (all classes)

Category : Uncategorized

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

NB: While strictly speaking, the term “mass” is more correct to use than the term “weight” (since mass is measured in kilograms and grams), in Operation Maths, we defer to using the term “weight” as that is the term used in the Primary Maths Curriculum (1999), as well as being the term most frequently used by the general population. To find out more about the difference between mass and weight, click here.

### Initial exploration – CPA approach

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

In the infant classes, this should occur through exploration, discussion, and use of appropriate vocabulary eg heavy/light, heavier than/lighter than, weighs more/less etc. The children should also be enabled to sort, compare and order objects according to weight.

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

• The children examine pairs of objects and make comparisons, e.g. lunchbox and schoolbag, chair and book, crayon and pencil case. Encourage the children to ‘weigh’ these objects in their hands; using outstretched hands, either to the side or in front of the body, as this can help the children get a better sense of which object is heavier/lighter.
• Elicit from the children how they might verify their hand-weighing. Introduce a balance and demonstrate how to use it. If sufficient balances are available allow one per group of four to six children. If there are not enough commercial balances, a simple alternative is to use a clothes hanger, from which two identical (ask the children why these need to be identical) baskets, trays or bags are hung (see video below).

• Move on to pairs of objects whose difference in weight may not be obvious, e.g. crayon and marker. Let individual children test pairs of objects on the balance.
• Examine pairs of objects where one is larger but lighter, (e.g. a big piece of paper and a stone, a ball of cotton wool and a pebble, a feather and a marble) and pairs of objects where the objects may have a similar size but different weights (eg a ping pong ball and a golf ball). These experiences enable the children to understand that weight is not related to size.
• This can then progress to incorporate a direct comparison of the weight of three or more objects, to now also include the labels heaviest/lightest. It is important at this stage that the children realise that if A is heavier than B and B is heavier than C, then, without further direct comparisons, we know that A is heavier than C, that A is the heaviest of all three and C is the lightest. This is a very important concept for the children to grasp.
• In a similar way, the children can estimate and record the weight of objects using non-standard units (e.g. cubes, marbles etc) and standard units of weight (e.g. a bag of sugar as a kilogram weight). Initially, when using standard units (e.g. kilogram) they will be recording the weight of objects as being heavier than/lighter than/the same weight as a kilogram.

HINT: 1/2kg and 1/4 kg weights for comparison can be made using that weight of rice, sand etc in ziploc bags. Challenge the children in 2nd class up to come up with ways to make these, and other, weights using only the balance (ie without using a scales). Making these weights could become an alternative homework task.

### Using scales: estimating & measuring

From Operation Maths 5, Pupils’ Book

As the children progress in their understanding of the concept of weight they will begin to appreciate the need for more accurate means to record weight, i.e. using a weighing scales. It is an advantage to have a wide selection of different types of scales available (including kitchen and bathroom, digital and dial) so that the children appreciate that not all scales are the same, and that their measuring skills have to be flexible enough to be able to adapt to the different types.

HINT: Some scales (eg luggage scales, etc) can often be purchased relatively cheaply from value shops. Alternatively, ask the children to bring in a scales from home to use in class while working on this topic.

As always, the children should be encouraged to estimate before measuring.  This can be done by hand-weighing and can incorporate the comparison of the weight of the unknown object with that of a known weight eg holding a lunch box and a bag of sugar in outstretched hands and estimating the weight of the lunchbox in kg and g based on how much heavier/lighter it feels in comparison to the 1kg weight.

Rather than estimating the weight of A, B, C and D before weighing A, B, C and D, it would be better if the children estimated the weight of A and then measured the weight of A, estimated the weight of B and then measured the weight 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 weight, and to use this sense to produce more accurate estimates.

When measuring weight using scales with dials, advise the children to first examine the markings to identify the major makings and to calculate the measure of the minor makings/intervals. When appropriate to the type of scales, encourage the children to read the scales at eye level to obtain a more accurate reading. For demonstrations purposes, a large interactive scales such as the one here, could be used

When the children have experienced using a variety of scales they should then be afforded the opportunity to choose which instrument (and which standard unit) is most appropriate to measure the weight of various items. In this way, they 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 weight

From fourth class on, the children will be expected to rename units of weight, appropriate to their class level. While changing 1,250 g to 1kg 250g or 1.25 kg, will typically be done correctly, converting figures which require zero as a placeholder (eg 1 kg 50 g, 2.6 kg ) 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:

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

T-charts, one of the three key visual strategies for problem-solving used throughout Operation Maths, can be very useful when renaming units of weight, 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 weights 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.