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Digging Deeper into … Area (2nd to 6th)

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

When most of us think of area, we probably think of Area = Length x Width. And this in itself hints at the difficulties with this topic; our knowledge of area often centers around a formula rather than understanding the concept of area (and the ability to visualise area) as the amount of space that a surface covers/takes up (as defined in the Maths Dictionary for Kids).

Area is introduced in Operation Maths 2. Initially, the children are enabled to consider space on a surface and which has the greater area (covers more) or the lesser area (covers less) as shown below.

In Operation Maths:

  • Area is taught after 2-D Shapes as the children will need to use their knowledge of the properties of 2-D shapes and tessellating patterns to appreciate which shapes are best to accurately cover a surface.
  • Area is taught after Length as, from 4th class up, the children require previous experience of measuring the length of an object/figure.
  • Area is also taught after Length in 4th class up, so as to avoid the children meeting both area and perimeter, initially, at the same time. That said, once it appears that the children have grasped the concept of area as the size a surface covers, then the connections with perimeter should be explored (see more on this below)
  • Because, in Operation Maths 6, the chapter on Area conveniently follows on from the chapter on Length, this also allows children to measure/calculate areas on room plans using their knowledge of scale, introduced in the Length chapter. This can be extended by the children measuring the dimensions of a specified area in the school grounds, e.g. pitch, car park and drawing a plan of the area to different scales.

Measuring area

Measuring area means to establish the area of a shape by measuring and/or counting the number of square units required to cover it (or it covers, when laid on top). Initially in second class, and as revision in third class, the children will be exploring this using non-standard units that are both square and non-square, for example playing cards, envelopes, etc. Through this exploration, it is hoped that the children will come to the realisation that it is preferable to use a standard square unit.

At this initial measuring phase, the children should be given as many opportunities as possible to measure the area of both regular and irregular shapes. These experiences could include:

  • Making shapes on a geoboard with elastic bands and measuring the area within; this can be modeled also on this online interactive geoboard. 
  • Placing transparent/translucent shapes on a grid to count the square units covered by the shape. Progress to using opaque shapes, as these are more challenging. The Operation Maths Sorting eManipulative can also be used to model this (see image below)
  • Make shapes that have the same area but look different. To do this, give the children  opportunities to draw different shapes of equal area on squared paper; “same area value, different appearance”. Again, this can be modeled, as shown below, using the Operation Maths Sorting eManipulative.
  • In the senior classes, square tiles, unifix cubes and/or the units in base ten blocks  can be used to link the concept of “same area value, different appearance” to both the area model of multiplication and identifying the various factor pairs for a number as shown in Number Theory. For example, the children can make rectangles of various dimensions, but all with an area of 36, and thus they can identify that the the factors of 36 are 1 x 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6.

In Operation Maths 3, by using squared paper/grids the children are introduced to using a standard square unit for measuring area. If the squared paper/grids are also centimeter grids this leads logically on to work in 4th class, where this square unit is then identified specifically as a square centimetre.

Estimation and efficiency

When using both non-standard and, later, standard square units, the children should always be encouraged to estimate the area first before measuring. As mentioned previously in the post on Length, rather than estimating the area of A, B, C and D before measuring A, B, C and D, it would be better if the children estimated the area of A and then measured/counted the area of A, estimated the area of B and then measured/counted the area 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. In this way, the children will also begin to develop their sense of space.

Some shapes may cover only parts of squares and this allows for opportunities to discuss what strategy to use to count these, for example two half squares count as one, less than half a square does not count, more than half a square counts as one.

As the children’s understanding develops, they should also be encouraged to come up with increasingly more efficient strategies for measuring area:

  • “How did you find out the area of the rectangle?”
  • “Did you count the squares?”
  • “Is there a faster (more efficient way) to count the squares rather than counting them in ones? Explain. “

Allow the children to verbalise and explain their strategies, as this discussion will likely reveal approaches that incorporate aspects of repeated addition and/or multiplication, thus leading on well to the children deducing a method to calculate area.

Calculating area

The children begin to calculate area as opposed to measuring (counting area) in 5th  class. However, this should not be introduced purely with the introduction of the formula for calculating the area of a rectangle, rather, as mentioned above, it is hoped that though sufficient opportunities of counting squares in previous classes that the children will now suggest more efficient strategies, including repeated addition and multiplying the length by the width. Considering also, that Operation Maths regularly uses the visual image of rectangular arrays to model multiplication (referred to as the area model), these experiences in multiplication will prepare the children well for the concept of calculating area via multiplication.

Initially, it is preferable that the children are calculating the area of shapes that can be easily checked by measuring. Then, when ready, they should progress to calculating area using more abstract measures such as millimeters, ares and large numbers of metres. They can also apply their knowledge to calculating the area of other shapes (eg triangles) and to irregular shapes that can be easily partitioned into rectangles (often referred to as compound shapes).  Finding the area of a circle (6th class) is by counting squares only and is covered in the chapter on the Circle.

Area and perimeter

As mentioned above, to avoid confusion between the concepts of area and perimeter, it is important that they are both taught separately, initially. The concept of perimeter as the length around the outside of a shape is not introduced until 4th class, meaning that in 2nd class and 3rd class the children can just explore the concept of area, without the confusion of adding perimeter to the mix!

When ready, the children can begin to explore the connections between the two concepts. And it is essential that both concepts are taught, using a visual context e.g.:

  • fences (perimeter) and sheep/grass (area)
  • skirting boards (perimeter) and tiles/carpet (area)
  • fences (perimeter) and stone slabs (area)
  • or any other context with which the children might be most familiar (see also the video at the end with shows the both concepts in various contexts)

The children can build models and/or draw outlines to represent area and perimeter:

  • make a fence using lollipop sticks or match sticks on large sheets of paper and sketch the square units within the border to match the length of each unit of “fence”.
  • Place the units from base ten blocks on a centrimetre square grid as sheep and draw units of fencing around them. Or use unifix cubes to do the same but on 2cm square grids as unifix cubes are 2cm long on each side.

Through this exploration, it is likely that the children will begin to realise that the perimeter of a rectangular shape does not determine the area of the shape. Using the concrete materials allow the children to construct both rectangles of constant area but varying perimeter and rectangles of constant perimeter with varying areas and to help develop the concept. Again, use a context if possible to reinforce the two concepts:

  • A farmer wants to build a sheep enclosure for 12 sheep, giving each sheep one square unit of space (use base ten units or cubes, as shown below). Show three different ways this could be done. Which way requires the most fencing? Which requires the least fencing?
  • A different farmer has 24 units of identical fencing. Show three different ways the fencing could be arranged. Which arrangement can take the most sheep, giving each sheep one square unit of space? Which arrangement can take the least sheep?

Some of the children may discover that the most efficient use of fencing, to produce the largest area, will be a square shape or a shape closest to a square, if not possible to make a square. In a similar way, in 6th class, when the children begin to investigate surface area, the children can investigate how the volume of a shape does not determine the surface area of the shape. They could use the base ten units (or any other available cubes) to build cubes/cuboids with the same volume (eg 12, 18, 24 etc cubic units), but in different arrangements each time, and measure (count)/calculate the surface area of each resulting arrangement.

Further Reading and Resources:

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

In Operation Maths for 5th Class the children are formally introduced to the concept of directed numbers (from the strand of algebra) and are enabled to:

  • Explore and identify directed numbers, i.e. numbers above and below zero, including zero
  • Describe and record directed numbers as positive or negative
  • Use directed numbers to represent real-life situations
  • Compare and order directed numbers
  • Solve problems involving directed numbers.

In Operation Maths for 6th class, this knowledge and understanding is revised and extended to include the addition of positive and negative numbers.

Directed numbers (formally known as integers, the set of positive and negative whole numbers including zero) can be a very difficult concept for some children for a number of reasons:

  • Being able to explore numbers less than zero requires a solid understanding of zero, an abstract concept in itself since it indicates the absence of something
  • Many children find it difficult to comprehend how something can be less than zero. Therefore, when ordering it can be difficult for many children to appreciate that 0 is greater than -1, -2, etc.
  • Operations involving integers are particularly troublesome as having similar signs so close together can be very confusing.

Use real-life contexts

In fifth class, since this is the first time that the children have been formally introduced to negative numbers, it is essential that this happens through reference to real-life contexts, e.g. temperature, buttons in a lift, goal difference in soccer league tables, depths in a swimming pool, or any other context with which the children can readily identify. Other real-life contexts such as bank account balances, elevations, etc., can also be used.

 

Use ‘positive’ and ‘negative’ rather than ‘plus’ and ‘minus’

Whilst real-life contexts are useful, they can also be confusing, particularly when plus and minus are used to describe directed numbers. Insist that the children use the language of positive and negative, since that is more correct mathematically. Using positive and negative can also reduce complications in future classes when calculating with directed numbers, e.g. avoiding having to say three minus minus five for 3-(-5). This is particularly important when the children progress to adding positive and negative numbers: for example (–3) + (+9) should be read as ‘negative three add/plus positive nine’ rather than ‘minus 3 plus plus 9’. It is also important that the children recognise that positive numbers can be written either with, or without, the positive sign, therefore we can assume that any number without a sign is positive.

To further reduce complications with written forms of directed numbers, raised signs should be used for the positive and negative (as shown in image below) and brackets should be used to make it easier for the child to distinguish between the operations signs (i.e. + and -) and the directed numbers signs.

 

Representing directed numbers

Since negative numbers are so abstract, it is vital that opportunities to represent them concretely and pictorially are maximised. Many children consider only the cardinality of numbers, i.e. that they represent the number of objects in a set, whereas understanding of directed numbers relies on understanding the ordinality of numbers; this is why visual representations are so vital.

One way to do this, is to use a counting stick, if available. Start with the counting stick in a vertical position, asking the children that if the centre line is now representing zero, can they identify the points above and below that?

  • Initially describe the positions as “one above zero”, “two below zero” etc
  • Then, when appropriate, introduce positions as “positive three”, “negative four” etc
  • Development: if each interval now represents two/ten/five as opposed to one, how does this change the identity of the positions?
  • Finally, move the counting stick into a horizontal position, so that the negative numbers are now to the left of the centre/zero and the positive to the right (as the children see it).

Operation Maths users can also use the counting stick eManipulative to model the counting stick in a horizontal position, as shown below (Hint: zoom the screen in to 150% to view the numbers better).

Number lines are also very useful. When using number lines (or a counting stick), emphasize that numbers get smaller in value as you move above/to the left of zero, and larger as you move below/to the right of zero . If possible, have a number line that includes negative numbers on permanent display in the classroom.

The Operation Maths MWBs can also be used to create a dynamic number line for ordering activities. For example, ask individual children to write a specific integer on their MWB. Then ask a group of children, each with different integers,  to stand out and put their MWBs in order. These numbers could even be hung from a ‘washing line’ using clothes pegs.

In sixth class, to better illustrate the processes involved with the addition of integers, teachers are encouraged to use the positive and negative chips, that accompany the sixth class books as part of the free ancillary resources (they’re on the same sheet as the images of the base ten block for modelling decimal numbers).

The teacher can also use the positive and negative chips built into the Sorting eManipulative to model calculations on the class IWB, as shown below (the positive and negative chips are located at the end of the numbers tab).

The children can then progress to using number lines to both reinforce their understanding of addition with integers and extend it to bigger numbers. A physical number line that the children can walk along would be ideal initially, as this suits the kinaesthetic learners and the bigger nature makes it easier for teachers to assess the children’s ability to use the number line before they move to smaller scale number lines in books and copies.

 

Further Reading and Resources

  • Dear Family, your Operation Maths Guide to Directed Numbers 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 the Algebra concepts, including directed numbers.
  • Operation Maths users don’t forget to use the Counting Stick eManipulative as mentioned earlier and to check out the Maths Around Us video for this topic on Edco Learning. Check out the first page of the Directed Numbers chapter in the Pupils Books for a quick synopsis of the suggested digital resources and then refer to the Directed Numbers chapter in the TRB for more detailed information.
  • NRICH: selection of problems, articles and games for negative numbers. In particular, check out Number Lines in Disguise.
  • Check out this Pinterest Board of Algebra ideas

Digging Deeper into … Length (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 counting and numeration, please check out the following post: Dear Family, your Operation Maths Guide to Length.

Initial exploration of Length

Initial experiences of length in the infant classes should occur through exploration, discussion, and use of appropriate vocabulary eg long/short, tall/short, wide/narrow, longer, shorter, wider than etc. The children should also be enabled to sort, compare and order objects according to length or height.

The Aistear Play suggestions in the Operation Maths TRBs for Junior and Senior Infants (see example above) provide very useful ideas that can be used as the basis for purposeful exploration and discussion, for example, in the case of the garden centre-themed suggestions above:

  • What plant/tree is taller/shorter?
  • What gardening tool is longer/shorter?
  • What plant pot/row of plants is wider/narrower?

Concrete-based exploration should follow and initial questions and discussion, to include direct comparison of the length of two objects, labelling them as long/short and/or longer than/shorter than etc. This can then progress to incorporate a direct comparison of the length of three or more objects, to now also include the labels longest/shortest. It is important at this stage that the children realise that if A is shorter than B and B is shorter than C, then, without further direct comparisons, we know that A is shorter than C, that A is the shortest of all three and C is the longest. This is a very important concept for the children to grasp.

 

Non-standard and standard units

Children in senior infants should begin to estimate and measure length using non-standard units of measure, such as lollipop sticks, straws, pencils etc. Non-standard units are specifically chosen, as opposed to the standard measures of metres and centimetres, as most objects that the children will choose to measure will be longer than 10cm, therefore outside the number limit of senior infants. Furthermore, the metre is almost too big a unit for them to work with at this stage, since the heights of many children in senior infants would be less than this. It’s difficult to develop a sense of a metre when it’s bigger than yourself! Whereas lollipops, straws etc are not too big nor too small for their hands and easier to work with. The children could also use a whole ruler as a non-standard unit in itself e.g. “the table is three rulers long”.

When the children have had some experiences measuring they should then be afforded the opportunity to choose which instrument is most appropriate to measure the length/height 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.

Children in first and second classes should also be afforded the opportunity to explore non-standard units of length prior to being introduced to the standard units. Historical non-standard units can be introduced also eg spans, digits, cubits, strides etc. Work using these measures will not only encourage the children to appreciate the need for standard units, but once they are introduced to the metre (first class) and the centimetre (second class) they should also try to identify which non-standard units are closest to the standard units, eg the child’s stride or arm span is often close to a metre and their digit is close to a centimetre (quick investigation for second class: the width of which of your fingers is closest to a centimetre?). Connecting these standard units of measure back to the children themselves helps them identify with the measure and helps them internalise a sense of its size.

 

Estimating & measuring

The children should always be encouraged to estimate before measuring. And rather than estimating the length of A, B, C and D before measuring A, B, C and D, it would be better if the children estimated the length of A and then measured the length of A,  estimated the length of B and then measured the length 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.  As mentioned previously, the child’s ability to connect standard units of length to themselves, not only helps them internalise a sense of length, but to also use this sense to produce more accurate estimates.

While the children begin using cm rulers in second class to measure length, they can still struggle to measure objects accurately in both this class and higher classes. Videos, such as the one below can be a useful way to demonstrate this skill to the whole class.

When children are performing tasks that require them to measure longer lengths (eg of the blackboard, the length of the room etc) using rulers or metre sticks, remind them to make sure that there are no gaps between their measuring instruments, that they keep them in a straight line and that they do not overlap. It can also be useful, if available, to use two metre sticks/rulers together so that once the second one is placed at the end of the first, the first ruler can then be moved to the end of the second ruler and so on.

 

Renaming units of length

From third class on, the children will be expected to rename units of length appropriate to their class level. While changing 136 cm to 1m 36cm or 1.36m will typically be done correctly, converting figures which require zero as a placeholder (eg 1m 5cm, 2.4m ) are quite often 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 and pictorial experiences as a way of checking the reasonableness of their answers, eg:

  • “2.4m equals 240cm because 1m is 100cm (point to an actual metre stick) and then 2m is 200 cm and .4 is nearly .5 which is half of a metre which is almost 50 cm”
  • “1m 5cm…well 1m is 100cm and then there’s 5cm more so it’s 100 plus 5, which is 105cm.

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

 

Perimeter and area

In Operation Maths, perimeter is deliberately taught separately from area; these topics are better taught independently because children can often confuse the concepts and processes of measuring area and perimeter.

Rather than introducing perimeter via a formula, it is critical initially that the children understand perimeter as the distance around the edge of a 2D shape and that they should actually measure around various 2D shapes as a way to identify the perimeter. Then, as they begin to look for more efficient ways to do this, they should be encouraged to discover and explain how they might calculate the perimeter of various types of shapes.

 

Scale drawings

Scale drawing is introduced in sixth class. Encourage the children to experiment with drawing rooms, houses or gardens to scale. This may be integrated with map reading or making maps in geography.

T-charts, once again, can be very useful when exploring scale. For example,  see how a t-chart can be used to help answer the question, below:

 

When the children are making scale drawings themselves, or when they are solving problems such as those above,  it can be useful to use a t-chart to set out some benchmarks measures for reference:

A fun way to extend scale drawings, and to integrate maths with visual arts, is to do scale drawings of a simple cartoon or line drawing, with the finished scale being larger than the original eg 1:4. This slideplayer presentation could be used for inspiration.

Further Reading and Resources:

Digging Deeper into … Decimals and Percentages (3rd – 6th)

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

In Operation Maths, decimals sits as a chapter of its own in third and fourth classes (allowing for specific time to focus on this new concept) and as a chapter combined with fractions and percentages in 5th and 6th classes. However, it is worth noting that the children would have informally encountered decimals since being introduced to euro and cent in first class. And, since decimals are inherently linked with both fractions and the place value system*, these topics, also, were encountered initially in first class, and thus the children’s understanding of decimals and percentages in the senior classes builds on their understanding of these related concepts. Indeed, in Operation Maths 5 and 6, the place value chapters include both whole and decimal numbers, since both are part of our base-ten system.

HINT: Read the related Operation Maths blog posts on fractions and place value at the links given.

 

Representing decimals and percentages

In keeping with the CPA approach used throughout Operation Maths, the initial introductory activities concentrate on experiences with concrete materials (e.g. straws and money, as shown below) and pictorial activities (e.g. colouring in fractions of shapes, dividing shapes and completing number lines) before progressing to abstract questions, where the focus is primarily on numbers and/or digits.  Concrete and pictorial-based experiences also promote the development of number sense and visualisation skills, allowing the children to become more adept at converting between various forms and, ultimately, being able to order, compare and calculate with fractions, decimals and percentages more efficiently and accurately.

In third class, the children encounter straws, place value discs, euro coins and blocks as examples of base-ten materials to represent hundreds, tens and units. Since, to represent one tenth, it is necessary to be able to fraction a unit, the only suitable base-materials to use are straws and ten cent coins (ie one tenth of a euro). Cutting up a straw into tenths really helps to demonstrate that one tenth is a very small part or bit of the whole. Operation Maths users can also use the Place Value eManipulative, accessible via Edco Learning to show various decimal numbers on the class IWB (see below).

In fourth class, with the introduction of the hundredth, the concrete materials become more limited, and one cent coins are used to represent hundredths of a euro (see below). Again, the Place Value eManipulative can also be used to represent hundredths; just select the Money HTU.th option from the drop down menu.

 

NB: With rounding of prices to the nearest 5c, we are likely to see fewer 1c and 2c coins in circulation, although recognition and knowledge of 1c and 2c coins are still part of the curriculum. Therefore, it was decided to continue to include them throughout the Operation Maths series for both this reason and because they are valuable for teaching purposes, especially when teaching decimals, as shown above. 

By fifth class, with the introduction of thousandths, the concrete possibilities have become even more limited; there is no coin to represent one thousandth of a euro because it is such a small and insignificant amount.  To this end, it it necessary to re-use base ten blocks, but with new values assigned to each block type (see below). It is very important that this is emphasized to the children,  and that they understand that the block that was previously used to represent a unit in 3rd and 4th class, is now being used to represent a thousandth in 5th and 6th class, simply because there are no other options!

Types of traditional representations for decimal numbers that have deliberately not been used throughout the Operation Maths series are place value abaci and dot notation boards. As mentioned previously in the post on Place Value, tasks which just involve the children identifying the number of dots on a notation board, or the number of beads on a place value abacus have not been included as they are not good indicators of a child’s understanding; rather they are simply demonstrating their number knowledge of numbers and digits from 0–9.

With the introduction of percentages in fifth class, since percentages are directly related to hundredths, all of the concrete resources that can represent tenths and hundredths can be used again to represent a percentage. When exploring how percentages relate to fractions and decimals, multiple hundredths squares can be used to encourage the visualisation of the various fractions, as in done in the Operation Maths Discovery book for fifth class (see below).

 

‘Same value, different appearance’

Because of the close connections between decimals, percentages and fractions, it is very important from the beginning that decimals (or to be more exact, decimal fractions) and percentages (from their introduction in 5th class), are taught as connected concepts with fractions (e.g. 1/2 = 0.5 = 50%,  1/4 = 0.25 = 25%) and that the children are encouraged to recognise them as different forms of an equivalent value i.e. ‘same value, different appearance’. Being able to convert into equivalent forms becomes very important when it comes to ordering parts of a whole that are expressed in various forms (see below).

Similarly, the children should be encouraged from the beginning to use both decimal language and fractional language when verbalising decimal notation, i.e. expressing 7.38 as ‘seven point three eight’ and also as ‘seven and thirty-eight hundredths’*. Using fractional language to read decimals reinforces the value of the digit(s) in the decimal place(s).

*However, when using decimal language, it is mathematically incorrect to say ‘seven point thirty-eight’, as the suffix -ty means tens.

Furthermore, the same decimal value can be written in various ways eg one-tenth can be written as 0.1, .1, 0.10, 0.100, etc. Many teachers often use only one form, usually 0.1, fearing that a variety of ways may confuse children. Conversely, using a variety of ways can actually help reinforce children’s understanding that all the above forms show one-tenth, with all forms (excluding .1) including unnecessary zeros.*
*Zeros can be necessary or unnecessary. In 30, the zero is necessary as without it, the value would be 3 units. In 0.3, the zero is unnecessary as without it, the value is still three-tenths.

 

Calculations with decimals and percentages

Addition and subtraction involving decimals can often appear to be mastered, until the children start using decimals of differing lengths (sometimes referred to as ragged decimals), and subsequent errors at this stage can reveal gaps in the children’s understanding of the concept. Remind the children regularly to lay out calculations so that the decimal places and decimal points are in line, and also use concrete materials (e.g. €1, 10c and 1c coins shown earlier) to encourage them to visualise the component parts of the number and how these similar parts must be added or subtracted accordingly e.g. for 1.24 + 2.3,  ‘4 hundredths and zero hundredths equals 4 hundredths, 2 tenths and 3 tenths equals 5 tenths’ etc.

Similarly, multiplication with decimals can also reveal gaps in understanding e.g. 0.4 × 3 might be answered as 0.12; again, where possible use concrete materials (e.g. use the tenths of the straws) to encourage the children to visualise the numbers i.e. ‘4 tenths times 3  (or three groups of four tenths) is 4 tenths plus 4 tenths plus 4 tenths, which is 12 tenths’; if straws are used we can see how that is equal to one whole straw and 2 tenths i.e. 1.2. Therefore, if interpreting multiplication as “group(s) of”, it is important that the children appreciate that 0.5 x 3 can be thought of as 1/2 group of 3 or as 3 groups of 1/2, both of which equal 1 1/2; this is modeled below using the Operation Maths Bar Model eManipulative, accessible on edcolearning.ie.

Another way to think about multiplications is as “rows of” and this concept of row leads very logically to the area model of multiplication. Base ten blocks are a very useful concrete material that can be used to represent multiplication with decimals via the area model, however it is important that the children appreciate that we are giving new values to each block ie each flat represents 1, the rods each represent 0.1/one tenth and the small cubes each represent 0.01/one hundredth. For more on how to use the area model of multiplications with decimals, please read on here: https://www.mathcoachscorner.com/2015/09/multiplying-decimals/

 

The cartoon above clearly illustrates how fostering number sense is as important as teaching procedures:
2.95 is nearly 3
So 2.95 x 3.2 is roughly 3 x 3 which is 9.something
The decimal point therefore goes in after 9

 

When solving a problem involving percentages, it is generally more efficient to convert the percentage to an equivalent fraction or decimal, and then solve the problem using a fraction or decimal approach; the decimal approach is often the most efficient  to use in a situation where a calculator is available/allowed. When using a fraction approach the children should be encouraged to use bar models to represent the quantities involved (see below). 

All activities should emphasise understanding and not just a procedural approach.
While it is important that children in sixth class are shown ways to calculate with more complex numbers, it is vital that this is done in such a way that the children begin to understand the purpose of the approaches and how they work (see below). Being told to just ‘multiply by 100/1’ or ‘divide by the bottom, multiply by the top’, does little to enable the child to understand the concept better.

 

Incorrect assumptions & misconceptions

As discussed earlier, while fractions, decimals, percentages and place value understanding are inherently connected, children may apply whole number and fractional understanding to decimals and percentages in such a way as to make incorrect assumptions.

For example:

  • A child may incorrectly assume that a number with more digits is bigger than a number with fewer digits, i.e. assuming incorrectly that 2.1 < 1.35
  • In fractions, as the denominator increases the fraction parts themselves get smaller (e.g. 1/2 > 1/10). Therefore, a child may assume, also incorrectly, that a decimal number with more digits is smaller, e.g. 1.32 < 1.2.
  • If 1/4 is greater than 1/8 , then a child may assume that 1.4 is greater than 1.8.
  • A child may convert a decimal value directly to a percentage (and vice versa), without changing their value, e.g. incorrectly assuming 0.15 = 0.15%.

There can be many other commonly held misconceptions and errors, including a child incorrectly thinking that:

  • Fractions, decimals and percentages are only parts of shapes and not numbers in their own right.
  • Percentages don’t go above 100% and fractions are never greater than 1.
  • A fraction such as 3/4 is only 3 groups of 1/4 without recognition that it can also be a 1/4  of 3; this is essential to understanding how a fraction can be converted into a decimal.
  • To calculate 20% of an amount, you divide by 20 (since to calculate 10% you divide by 10).
  • 0.2 equals a half or 12% is one twelfth, etc.

Again, many of these misconceptions and incorrect assumptions can be avoided by using a CPA approach to the teaching of this topic, with an emphasis on understanding rather than just doing.

Further Reading and Resources

  • Dear Family, your Operation Maths Guide to Decimals and Percentages includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
  • Virtual Maths Manipulatives for Fractions, Decimals and Percentages: Lots of tools that can be used in many different ways to explore these concepts.
  • Operation Maths users don’t forget to check out the extensive digital resources available for this topic on Edco Learning. These include Maths Around Us and Write, Hide, Show videos, Ready to go activities and Create Activities using the place value eManipulative, fractions eManipulative, 100 square eManipulative and bar modelling eManipulative. Operation Maths users in 3rd to 6th should check out the first page of the chapter in their Pupils Books for a quick synopsis of the suggested digital resources and then refer to their TRB for more detailed information.
  • Place Value, Decimals and Percentages Manual from PDST
  • NRICH: selection of problems, articles and games for decimals and percentages
  • Check out this Pinterest Board for further ideas for decimals and percentages as well as fractions.
  • Does the decimal point move? A one minute video which shows multiplying/dividing by 10/100 etc using the moving digits approach
  • Watch this video showing a teacher revising tenths and introducing hundredths

Digging Deeper into … Patterns and Sequences

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

This can be a difficult strand unit to track through the 1999 Primary Mathematics Curriculum; in junior and senior infants it is titled Extending patterns, in first and second classes it becomes Extending and using patterns, in third and fourth it is called Number patterns and sequences,  and in fifth and sixth classes it morphs into Rules and properties. However, it is always from the strand Algebra (check out the maths curriculum glance cards here for more detail) and a summary of the objectives reveals how pattern is at the heart of the strand unit at every class level:

Junior  Infants to Second Class > Algebra > Extending patterns >

  • identify, copy and extend patterns (colour, shape, size, number)
  • recognise patterns (including odd and even numbers;  predict subsequent numbers)
  • explore and use patterns in addition facts (1st & 2nd)

Third & Fourth Class > Algebra > Number patterns and sequences >
• explore, recognise and record patterns in number
• explore, extend and describe (explain rule for) sequences
• use patterns as an aid in the memorisation of number facts

Fifth & Sixth Class > Algebra > Rules and properties >
• identify relationships and record verbal and simple symbolic rules for number patterns

Different types of patterns

So then, are patterns and sequences the same thing? Actually, there are two main types of patterns:

  • Repeating patterns: repetitions of symbols, shapes, numbers etc., that recur in a specific way.
  • Increasing (growing) and decreasing (shrinking) patterns: An ordered set of shapes or numbers that are arranged according to a rule. Typically, sequence is also used to describe an increasing or decreasing pattern, particularly if it is a pattern of numbers.

Repeating Patterns

A repeating pattern should have a clearly identifiable core, i.e. the shortest sequence that repeats. It is a good idea to use the terminology of “core” right from the infant classes so that the children understand what is being asked of them.

Children can often copy patterns without even recognising or identifying the core. However, to become competent in accurately extending repeating patterns, it is vital to identify the core. Ways that the children can become more adept at this include verbalising the pattern out loud (“red, blue, yellow, red, blue, yellow, …”) and/or using concrete materials to model the pattern (see below); in this way it is easier to identify the core of the pattern by breaking it apart and laying it alongside the subsequent parts of the pattern to ensure that they match. This strategy of breaking and matching can also be used to help children check have they extended the pattern correctly. It is also for this reason that cubes and links can often be the easiest concrete materials to use for replicating and extending patterns and are preferable initially to threading beads, pegs on a pegboard etc . These can be used instead when the children are ready to progress to more challenging tasks.

        

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Sequences

Unlike repeating patterns, sequences are more linear; they tend to increase or decrease in specific ways; thus they are also referred to as increasing and decreasing patterns. The way in which the terms (the individual parts of the sequence) are ordered is governed by a rule. Similar to repeating patterns, the children need to be able to identify this rule in order to extend the sequence. To help the children identify the rule in numerical sequences, they should be encouraged to examine each given term and identify what has happened between it and the next term i.e. did the numbers increase, decrease, by how much etc.? They should then record this (e.g. +2, –3) below and between the terms, as modelled in the Operation Maths Pupil books and Discovery books.

Even when extending sequences in their copies, the children should be encouraged to leave an empty line below the sequences, to allow space for them to write in the differences between the terms.

Once the children are comfortable with sequences that increase/decrease by the same amount each term, they can progress to sequences that increase/decrease in varying but repetitive amounts, e.g. +2, +3, +2, …

Odd and even numbers

Odd and even numbers are an example of an increasing pattern/sequence, as the difference between each term is +2. Most children will find it simple enough to recognise odd and even numbers; typically they will tell you that if a number ends in 2, 4, 6, 8 or zero, is it even and, if it does not, it is odd. It is one thing to identify odd and even numbers in this way, but it’s another thing to visualise the numbers and appreciate why they are odd or even. Using concrete materials or pictorial representations is vital for the children to really develop their number sense
and their appreciation of how odd and even numbers interact.

As well as activities like that shown above, pairwise ten frames (ie ten frames placed vertically, as opposed to the more typical horizontal arrangement) can also be useful to model odd and even numbers. Such concrete or pictorial presentations can then be used to show how the total of an odd number and an odd number will always be even as the non-paired cube of both now join to form a pair. These activities also reinforce how only whole numbers can be classified as odd or even, even though a child may incorrectly assume that 1.2 is even, since it ends in 2.

Identifying patterns in addition and multiplication facts

Through concrete activities and activities using the 100 square, it is hoped that the children begin to appreciate the patterns in number facts and that that various groups of multiples are characterised by certain properties eg.:

  • When adding/subtracting 10 on the 100 square the answer is always the number directly below/above the starting number.
  • When adding/subtracting 9 on the 100 square the answer is always the number diagonally below left /diagonally above right the starting number.
  • The multiples of 10 always end in zero
  • The digits in the multiples of 9 always total 9 or a multiple of 9 (e.g. 9 × 11 = 99 and 9 + 9 = 18), etc.

Not only will knowledge like this greatly aid their ability to identify and recall the basic number facts, but it will also improve their ability to check and identify errors in their own, and others’ work (e.g. ‘173 × 5 = 858 … hmm, that can’t be correct because multiples of 5 should always end in 5 or 0; I need to do that again’).

Using T-charts to organise information

In the senior classes, when the rules governing the sequences are becoming more complex and less obvious, T-charts can be a very useful way to organise and present  the information. They can be particularly useful to help highlight the patterns and how these patterns are developing (see below). They can also provide the children with a clear way to explain how they see the rule. Therefore, the children should be encouraged to use them as a problem-solving strategy as much as is suitable.

Further Reading and Resources:

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

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