# Dear Family, your Operation Maths guide to Counting and Numeration

## Dear Family, your Operation Maths guide to Counting and Numeration

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Dear Family, given below is a brief guide to understanding the topic of counting and numeration, as well as some practical suggestions as to how you might support your children’s understanding at home. Also below, are a series of links to digital resources that will help both the children, and you, learn more about counting and numeration. The digital resources are organised according to approximate class level:

#### Understanding Counting and Numeration

Counting and numeration is about the counting words we use to tell the amount in a group, and the numbers we write for those counting words. And counting is not just about chanting a series of numbers …’one, two, three, four, five…’. It is about using these numbers with meaning, for example, understanding that the word ‘five’ can be written as 5, that it comes after four and comes before six, and that it can be used to describe the amount in a group of five items (and not just the label for the last item in the count). So, even though your child may know how to count to 10 or 20 or more, by the time they come to primary school, this does not necessarily mean that they understand the meaning of each number, or its place in the counting sequence. This is often described as the difference between rote counting (chanting a sequence of numbers) and rational counting (counting with understanding); for more on this, please check out the one minute video below.

Counting and numeration is a strand unit in Primary Maths for children in junior infants to second class only. Children in the senior classes will still do a lot of counting and numeration activities, but mainly as part of the strand unit place value. In the younger classes, the type of learning activities are very similar at every class level; the main difference is that each class level will have different number limits. In school, we expect that most children should be able to read, write and use numbers:

• 0-5 by the end of junior infants, and be able to count to 10
• 0-10 by the end of senior infants, and be able to count to 20
• Up to 99 by the end of first class
• Up to 199 by the end of second class

That is not to say that you should limit your child to only counting up to the number limit for his/her class level. In reality, children will encounter much larger numbers in the real world, than they will encounter in their maths books, so feel free to include larger numbers when you meet them. But, bear in mind that, even if a child can read or say a large number, it doesn’t necessarily mean that they understand it.

#### Practical Suggestions for Supporting Children

• Count, count and count some more! Count out the plates at the table, count out sweets or treats, count steps as you go up and down stairs, count down the days left to a birthday; use every opportunity for your child to hear you count, and when ready involve them in the counting.
• Say rhymes and sing songs that involve numbers or counting, for example, One, Two Buckle my Shoe, Five Fat Sausages, Ten Green Bottles, etc.
• Watch Numberblocks and Numberjacks. Many of the episodes from these two award-wining series from the BBC are available on-line and may also be available on your TV if you have BBC.
• Develop counting skills though play:
• Play tea-time with the toys, where each toy gets one cup, one plate, one bun etc. No toy should have more than one, and no toy should have none. These activities help to reinforce the one-to-one correspondence required in counting correctly.
• Play counting games at home (for example throw items into a basket/box and count them as you throw) and ordering games, for example where you layout playing cards in order. Or guessing games, where you estimate (guess carefully) the number of items in a container, bag etc., and then count to check.
• Play board games where the child has to throw a dice, recognise the number of dots shown, and then move on a counter that number of places.
• Play games where each number in a sequence (e.g. 1 to 10; 45 to 55; 103 to 113) is written on piece of paper/card and placed face-down. The child must turn over every piece in turn and read aloud the number. Then, he/she should put the numbers in a line in order. Finally, you could play hide and seek: remove a number from the line and your child has to tell you the missing number.
• Play counting games on car journeys, e.g. each child in the car picks a colour and counts every car of that colour that they see or meet on the road. The winner is the person who hits the highest number before the driver’s patience wears out!
• When your child starts to write numbers, you will need to monitor their number formation very carefully; it it very important that they don’t get into the habit of writing a number incorrectly.
• Draw your child’s attention to numbers around your home and in the wider environment, e.g. numbers on signposts, car registrations, phone numbers, the number of pieces in a jigsaw, page numbers on catalogues, the numbers on houses or hotel rooms. When you spot a number, ask them to read it out.
• With older children, when you are talking about numbers be careful to use the correct language e.g. for 125 say ‘one hundred and twenty five’ not ‘one-two-five’
• It’s an unfortunate convention, but the way we talk about numbers every day, can often be mathematically incorrect and/or confusing. For example, when calling out a mobile number, that starts with 08….. we will likely say ‘oh eight‘…… Yet 0 is a digit called zero, whereas O (said as ‘oh’) is a letter of the alphabet and not a number at all! So, when verbalising numbers with zero, try to get into the habit of saying ‘zero’ instead of ‘oh’.
• Numbers that end in ‘-teen’ or ‘-ty’ can be difficult for some children. In particular, some children can have difficulty hearing the difference between numbers ending in ‘-teen’ and ‘-ty’ when they are spoken out loud, e.g. ‘fifty’ (50) sounds very like ‘fifteen’ (15) when spoken, yet their values are very different. Try to say these type of numbers clearly, and encourage your child to say them clearly also, so that they appreciate the difference between these similar-sounding numbers.
• For more help and tips, check out this parents’ resource Topmarks: Learning Numbers

#### Digital Resources for Junior and Senior Infants

Underwater Counting: Count the underwater sea creatures and choose the matching numeral. Has different levels: numbers up to 5, up to 10.

Teddy numbers: Learn to count by giving Teddy the correct number of buns. Has different levels: numbers up to 5, up to 10 and up to 15.

The Gingerbread Man Game: Counting, matching and ordering games with options for numbers up to 5, and up to 10.

Ladybird Spots: Counting, matching and ordering games with options for numbers up to 5, and up to 10.

Toys Counting Game: Place the correct number of toys on the shelf. Counting to 5, 10 and using number words.

Curious George Hide and Seek: Find the number word, the numeral and the matching number of creatures. Numbers up to 10.

Curious George Apple Picking: Pick the number that is missing from he sequence. Numbers up to 9.

Count the Yeti 1-10: Count the number of yetis and shoot the correct number at the top.

Helicopter Rescue: Find on the number path the direct number that the computer asks, or find the number in between two given numbers. Has different levels: numbers up to 10, up to 20, up to 30, up to 50 and up to 100.

Caterpillar Count: Count and collect the numbers in order up to 15, to watch the caterpillar change into a butterfly.

Treasure Hunt: Help the pirate find his lost treasure by clicking on the island that shows the correct number. Select ‘Find the biggest number’ option and then adjust to set the maximum number.

Chopper Squad: Find a number 1 more/less or 10 more/less than a given number. Has different levels: numbers up to 20, up to 30, up to 50 and up to 100.

Blast Off: In the Find a Number game (red labels) you are asked to find, from 3 options, the direct number that the computer asks, or find the number in between two given numbers. Has different levels: numbers 10-20,  10-30, 30-60, and 60 to 99.

Caterpillar Ordering: Choose between ordering (where you put the given numbers in order) or sequencing (where you complete the sequence with the correct numbers from those given).  Has various levels including 1-5, 1-10 and 1-20.

Count and match: Count the items and drag over the matching numeral (up to 10)

Number Matcher: Find the matching number and number word.

Happy Numbers Pre-Kindergarten: Work through the activities from Module 1, counting to 5 and/or Module 3, counting to 10. Alternatively, go to Kindergarten, Module 1, numbers to 10

I Know it! – Counting: Scroll to numbers + counting  + place value to do any of the activities.

Splash Learn – Counting Games: An assortment of games organised according to US grade levels; junior and senior infants should choose among the games for kindergarten level.

Counting: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum)

#### Digital Resources for First and Second Classes

Please note: The digital resources for first and second classes often overlap with the place value digital resources for these classes, as the skills are very related.

Maths Visuals – Counting by one: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

Maths Visuals – Counting above 100: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

Maths Visuals – Place Value Concepts: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

Helicopter Rescue: Find on the number path the direct number that the computer asks, or find the number in between two given numbers. Has different levels: numbers up to 10, up to 20, up to 30, up to 50 and up to 100.

Chopper Squad: Find a number 1 more/less or 10 more/less than a given number. Has different levels: numbers up to 20, up to 30, up to 50 and up to 100.

Blast Off: In the Find a Number game (red labels) you are asked to find, from 3 options, the direct number that the computer asks, or find the number in between two given numbers. Has different levels: numbers 10-20,  10-30, 30-60, and 60 to 99.

Treasure Hunt: Help the pirate find his lost treasure by clicking on the island that shows the correct number. Select ‘Find the biggest number’ option and then adjust to set the maximum number.

Caterpillar Ordering: Choose between ordering (where you put the given numbers in order) or sequencing (where you complete the sequence with the correct numbers from those given).  Has various levels including 1-100.

Coconut Ordering: Hit the numbers in order of size. Select ‘numbers’ and then choose from numbers up to 10, up to 20, up to 100 (in tens) or up to 100 (any number).

Battleship Numberline: Can you blow up the enemy submarines? This game starts very easy, where you must click the correct number on the number line, but then the game progresses in difficulty as the player must work out where a given number would be placed on the blank number line. Choose the whole number game.

Happy Numbers Kindergarten: First class could explore the activities from Module 5, Numbers 10 – 20 and Counting to 100.

I Know it! – Counting: Scroll to counting and number patterns to do any of the activities.

Splash Learn – Counting Games: An assortment of place value games organised according to US grade levels; first class should choose from among the games for first grade, and second class should choose from the games for first and second grade.

Counting: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum)

## Dear Family, your Operation Maths guide to Place Value

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Dear Family, given below is a brief guide to understanding the topic of place value as well as some practical suggestions as to how you might support your children’s understanding at home. Also below, are a series of links to digital resources that will help both the children, and you, learn more about place value. The digital resources are organised according to approximate class level:

#### Understanding Place Value

Place value is about exploring the base-ten number system we use: how our numbers are made up of digits, each of which represent different values, depending on their position or place in the number. In the senior classes the children will explore place value in numbers with a decimal point, as well as numbers without a decimal point (whole numbers).

No matter how large a number is, you really only need to know how to read a three-digit number, to be able to read any size number. This is because the digits are always organised in groups of three, as you can see in the image below. However, we do also need to know the significance of the commas. For example, three (3) million, six hundred and twenty-three (623) thousand, nine hundred and fifteen (915) = 3,623,915: the comma closest to the units is read as thousand, the next comma is read as million, etc.

At its most basic level, central to understanding our place value system, is to recognise that 10 single items or units or ones, can be grouped together to make a ten; that 10 tens can be grouped together to make a hundred; that 10 hundreds can be grouped together to make a thousand etc. In school, the children have lots of different materials that they can group together, or exchange, such as cubes, bundles of sticks, counters on ten frames and place value discs. At home, the children could bundle cotton buds or cocktail sticks or trading cards into groups of tens and fasten them with an elastic, or group identical pieces of lego into sticks of ten, or count out beads or buttons or pieces of pasta into small containers or bags as groups of tens.

In school, the type of place value learning experiences that the children have, are very similar at every class level; the main difference is that each class level will have different number limits. In school, we expect that by the end of first class, most children will understand place value in numbers up to 99, in second class up to 199, in third class up to 999, in fourth class up to 9,999, in fifth class up to 99,999 and in sixth class there is no limit … millions, billions, trillions even!

That is not to say that you should limit your child to the number limit for his/her class level. In reality, children will encounter much larger numbers in the real world, than they will encounter in their maths book, so feel free to throw bigger numbers at them. But, bear in mind that, even if a child can read or say a complicated number, it doesn’t necessarily mean that they understand its place value.

#### Practical Suggestions for Supporting Children

• Ask your child to read out loud any numbers they meet around your home and in the wider environment, e.g. numbers on signposts, car registrations, the number of pieces in a jigsaw, page numbers on catalogues, the numbers on houses or hotel rooms, larger numbers on fact books e.g. Guinness Book of World Records, recorded times for races, etc.
• Correct language: When you are talking about numbers be careful to use the correct language e.g. for 91,856 say ‘ninety one thousand, eight hundred and fifty six’ not ‘nine-one-eight-five-six’ and for 23.95 say ‘twenty three point nine five’.
• Zero does not equal ‘oh’! It’s an unfortunate convention, but the way we talk about numbers every day can often be mathematically incorrect and/or misleading. For example, when calling out a mobile number, that starts with 08….. we will likely say ‘oh eight‘…… Yet 0 is a digit called zero, whereas O or ‘oh’ is a letter of the alphabet and not a number at all! So, when verbalising numbers with zero, try to get into the habit of saying ‘zero’ instead of ‘oh’.
• Numbers that end in ‘-teen’ or ‘-ty’ can be difficult for some children. In particular, some children can have difficulty hearing the difference between numbers ending in ‘-teen’ and ‘-ty’ when they are spoken out loud, e.g. ‘fifty’ (50) sounds very like ‘fifteen’ (15) when spoken, yet their values are very different. Try to say these type of numbers clearly, and encourage your child to say them clearly also, so that they appreciate the difference between these similar-sounding numbers.
• Rounding large or awkward numbers is something we do to make them easier to say or report. For example, if there was 91,856 people at a concert or a match, the media might report that there was just over ninety thousand or there was almost ninety two thousand people in attendance. When you encounter numbers in the media, encourage your child to round them; ask him/her what the number would be roughly/approximately. If you come across a number that has already been rounded, together you could guess/speculate as to what the exact number might have been.
• Make place value fun!
• Play counting games on car journeys, e.g. each child in the car picks a colour and counts every car of that colour that they see or meet on the road. The winner is the person who hits the highest number before the driver’s patience wears out!
• Race to the page! Challenge your child to try to find certain page numbers, in books with plenty of pages, as quickly as they can. Use a dictionary or other reference book, or even an Argos catalogue and call out a page number, for example ‘three hundred and ninety’ and see how quickly that page can be found. If you have more than one copy of a suitable big book or catalogue, two players can race against each other.
• Play some simple place value games using dice or playing cards
• Play any of the online interactive games below

#### Digital Resources for First and Second Classes

Khan Academy – Intro to Place Value: this video and the videos that follow, explore place value in 2-digit numbers and then answer the practice questions. You can also register for a free Khan Academy account to record your progress and explore other areas of Grade 1 maths.

Happy Numbers – Place Value Activities: A series of lessons and activities; do activities from Module 4 and/or 6.

Maths Visuals – Counting by one: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

Maths Visuals – Counting above 100: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

Maths Visuals – Place Value Concepts: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

White Rose Place Value: a series of lessons on place value within 20. These lessons could be followed up with other place value lessons in year 1 or year 2

Place Value Grouping Video: Watch a video of how ones (units) can be grouped into tens, to make various numbers.

Candy Machine: Help make up the orders of candy sticks by using bundles of tens and ones

Dienes Penalty Shoot Out: Identify the number of counters and create numbers using Dienes blocks (aka Base Ten Blocks). Choose game mode to earn penalty chances, and then numbers up to 20, 50 or 100.

Place Value Basketball: Select the correct number to match the image. Work your way up through the various options/levels.

Lifeguards: Click and drag into the place value grid, the correct number of place value discs to make up the given number. Choose between 0-50 or 0-100 options. You can also play a similar game called Shark Numbers

Maths Goalie – Reading numbers:  Read the numbers in word form and then input the same number but in standard form. Choose reading numbers, and then number to 20 or 100.

Place Value Charts: Make a given number by combining the parts that make up the number. Select practice and then T O (Tens and Ones) in either column.

Rocket Rounding: A multiple choice game involving rounding numbers: start with rounding numbers up to 99 and with the easier option of having a number line and then try to play the other more difficult option, no number line.

Battleship Numberline: Can you blow up the enemy submarines? This game starts very easy, where you must click the correct number on the number line, but then the game progresses in difficulty as the player must work out where a given number would be placed on the blank number line. Choose the whole number game.

I Know it! Place Value: Scroll down to place value to do any of the activities. For children at the beginning of first class try Place Value up to 20, Base Ten blocks up to 20 and Count to 100 instead. There are some more advanced activities in the second grade section.

Place Value: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum)

#### Digital Resources for Third and Fourth Classes

Khan Academy – Place Value: in this video and the videos that follow, explore place value in 3-digit numbers and then answer the practice questions (says Grade 2, but is suitable for 3rd class). Fourth class student can access similar activities for 4-digit numbers and larger here. You can also register for a free Khan Academy account to record your progress and explore other topics.

Happy Numbers – Place Value Activities: A series of lessons and activities; for numbers up to 1,000 do Module 3. For rounding to the nearest ten and hundred do Module 2 here.

White Rose Place Value: a series of lessons on place value suitable for 3rd class. These lessons could be followed up with other place value lessons in year 3 or year 4

Place Value House: video lesson that explores hundreds, tens and ones (units), suitable for 3rd class.

Expanded Form: A video that introduces expanded form and explains how we can expand numbers to see the parts that make it, suitable for 3rd class.

Dienes Penalty Shoot Out: Identify the number of counters and create numbers using Dienes blocks (aka Base Ten Blocks). Choose game mode to earn penalty chances, and then numbers up to 1,000 or 5,000.

Place Value Basketball: Select the correct number to match the image. Work your way up through the various options/levels.

Lifeguards: Click and drag into the place value grid, the correct number of place value discs to make up the given number. Choose between 0-500 or 0-1,000 options. You can also play a similar game called Shark Numbers

Maths Goalie – Reading numbers:  Read the numbers in word form and then input the same number but in standard form. Choose reading numbers, and then number to 1,000 or 10,000.

Place Value Charts: Make a given number by combining the parts that make up the number. Select practice and then either H T O (for third class) or Th H T O (for fourth class) in either column.

Rocket Rounding: A multiple choice game involving rounding numbers to the nearest 10 or 100, up to 999 or 9,999. Start with the easier option of having a number line and then try to play the other more difficult option, no number line.

Battleship Numberline: Can you blow up the enemy submarines? This game starts very easy, where you must click the correct number on the number line, but then the game progresses in difficulty as the player must work out where a given number would be placed on the blank number line. Choose the whole number game.

I Know It! – Place Value: Scroll down to place value to do any of the activities with suitable number limits. There are some more advanced activities in the third grade section.

Place Value Games: An assortment of place value games using numbers of various sizes. Third class pupils should start with games up to 999 (three-digit numbers) and fourth class should start with games up to 9,999 (four-digit numbers)

That Quiz – Place Value: This quiz has lots of options, on the left hand side, that can be changed to suit the ability of the child. In place value, the lowest level is 3. Each time do the set 10 questions, if you get 10 or 9 correct, go up a level; if not stay at that level. There are lots of different types of activities: For Identification (it automatically starts on this) you must identify the value of certain digits; other options are conversions, rounding and sums.

Place Value: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum)

#### Digital Resources for Fifth and Sixth Classes

Khan Academy – Place Value: in this video and the videos that follow, learn about place value in larger numbers and then answer the practice questions. You can also access similar activities for decimal numbers here. If you register for a free Khan Academy account, you can record your progress and explore other topics.

Happy Numbers – Place Value: A series of interactive lessons and activities on numbers up to one million. Do Module 1 Topic A, B and C

White Rose Place Value: a series of lessons on place value suitable for 5th class. These lessons could be followed up with other place value lessons in year 5 or year 6

How big is a billion? It is very difficult to visualise the size of a million, or a billion, of anything. This video demonstrates the length of a thousand, a million, and a billion coins if they were placed top to bottom.

Maths Goalie – Reading numbers:  Read the numbers in word form and then input the same number but in standard form. Choose reading numbers, and then numbers to 1,000,000 or 10,000,000.

Place Value Charts: Make a given number by combining the parts that make up the number. Select practice and then either whole numbers or decimal numbers, in either column.

Rocket Rounding: A multiple choice game involving rounding numbers, using whole numbers or decimal numbers. Start with the easier option of having a number line and then try to play the other more difficult option, no number line.

Battleship Numberline: Can you blow up the enemy submarines? This game starts very easy, where you must click the correct number on the number line, but then the game progresses in difficulty as the player must work out where a given number would be placed on the blank number line. Choose the whole number or decimals game.

Who wants to be a Hundredaire? Game show-like quiz based on place value.

I Know It! – Place Value: Scroll down to place value to do any of the activities with suitable number limits. There are some more advanced activities in the fourth grade section.

Place Value Games: An assortment of place value games using numbers of various sizes. Fifth class pupils should start with games up to 99,999 (five-digit numbers) and sixth class should start with games above this. There are similar games based on decimal numbers accessible here.

That Quiz – Place Value: This quiz has lots of options, on the left hand side, that can be changed to suit the ability of the child. In place value, the lowest level is 3. Each time do the set 10 questions, if you get 10 or 9 correct, go up a level; if not stay at that level. There are lots of different types of activities: For Identification (it automatically starts on this) you must identify the value of certain digits; other options are conversions, rounding and sums. Sixth class pupils looking for a challenge could try scientific notation.

Place Value: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum)

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

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of 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”.

## Thinking Strategies for Multiplication and Division Number Facts

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### What are number facts?

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

Some of the big ideas about number facts:

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

### What are thinking strategies?

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

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

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

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

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

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

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

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

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

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

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

### Computational Fluency:

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

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

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

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

From Number Facts 4

### The Number Facts Series from Edco

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

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

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

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

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

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

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

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

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

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

### Relationship between addition and subtraction

As shown in the videos above, addition and subtraction are inverse operations; we can demonstrate addition by adding more to an existing amount; the reverse action would involve removing an amount, thus demonstrating subtraction as take away. In contrast to traditional maths schemes, which often have separate chapters for each of these operations, Operation Maths predominantly teaches addition and subtraction together, as related concepts. Teaching the operations in this way will encourage the children to begin to recognise the relationships between addition and subtraction.

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

### CPA Approach within a context

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

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

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

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

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

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

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

### The meaning of the equals sign

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

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

### Looking at more complex numbers

In first and second classes, once introduced to operations using two-digit numbers, children can often have tunnel vision (or column vision) regarding addition and subtraction calculations: they “do” the units, and then the tens, without really looking at the whole numbers or the processes involved.

One way in which you can encourage the children to look at and understand these operations better is by using a CPA approach. This means that the children’s initial experiences should involve groupable base ten concrete materials (e.g. bundling straws or lollipop sticks, ten-frames and counters, unifix or multi-link cubes arranges in sticks of ten, see below), where a ten can be physically decomposed  into ten units and vice versa, before moving on to pregrouped base ten materials (eg base ten blocks/Dienes blocks, base ten money and/or Operation Maths place value discs) which require a swap to exchange a ten for ten units and vice versa.

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

When exchanging tens and units (or tens and hundreds in second class), reinforce that a ten is also the same as 10 units, and that a hundred is the same as 10 tens and is the same as 100 units. The use of non-canonical arrangements of numbers (e.g. representing 145 as 1H 3T 15U or  14T 5U), as mentioned in Place Value, can also be very useful to children as they develop their ability to visualise the regrouping/renaming process. The Operation Maths Place Value eManipulative, accessible on edcolearning.ie,  is an excellent way to illustrate this and explore the operations in a visual way.

### Mental strategies are as important as written methods

In first and second classes, the traditional, written algorithms for addition and subtraction, i.e. the column methods, are important aspects of these operations. However, in real-life maths, mental calculations are often more relevant than written methods. Also, as mentioned previously, children can often have tunnel vision (or column vision) regarding addition and subtraction calculations; they ‘do’ the units, then the tens, without really looking at the entire numbers or the processes involved. Therefore, while the column method for addition and subtraction is an important aspect of this topic, equally important is the development of mental calculation skills, via a thinking strategies approach.

From Number Facts 1 & 2

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

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 … Early Mathematical Activities

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Category : Uncategorized

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

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

It includes the strand units of:

• Classifying
• Matching
• Comparing
• Ordering

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

### Sets

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

### Matching

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

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

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

### Classifying

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

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

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

### Comparing and Ordering

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

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

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

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

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

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

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 … Decimals and Percentages (3rd – 6th)

Category : Uncategorized

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.

• 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 … Fractions (1st – 6th)

Category : Uncategorized

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.

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.

• Dear Family, your Operation Maths Guide to Fractions 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, and create Create Activities using the Fraction eManipulative, sorting eManipulative and bar modelling eManipulative. Operation Maths users in 3rd to 6th should check out the first page of the Fractions chapter in their Pupils Books for a quick synopsis of the suggested digital resources and thne refer to their TRB for more detailed information.
• Check out this Pinterest Board of Fraction ideas
• Fractions Manual from PDST
• Fraction Models: an informative article with images and teaching ideas, from K-5 Math
• NRICH: selection of problems, articles and games for fractions
• Watch any of these informative videos:

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

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.

### Online Resources

• 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.
• That Quiz: Use the factors quiz to practice identifying prime and composite numbers, prime factors, HCF and LCM. Use the exponents quiz to practice roots and exponents. To find out more about the potential of That Quiz across all strands and subjects, please read on here.
• Using Lego to develop math concepts: Read this article to discover ways to use Lego to explore square numbers, arrays and factors

• 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 Theory Board on Pinterest

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

Category : Uncategorized

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

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

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

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

### CPA Approach

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

Using base ten blocks to demonstrate multiplication as an area array

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

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

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

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

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

### Language & Properties

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

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

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

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

### Thinking Strategies for the Basic Number Facts

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

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

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

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

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

### Mental strategies are as important as written methods

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

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

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

### Problem-solving strategies

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

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

• Dear Family, your Operation Maths Guide to Multiplication & Division includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• Operation Maths Digital Resources: As always don’t forget to access the linked digital activities on the digital version of the Pupil’s book, available on edcolearning.ie. Tip: look at the footer on the first page of each chapter in the pupil’s book to get a synopsis of what digital resources are available/suggested to use with that particular chapter.
• For more hints and tips specific to each class level, check out the “What to look out for” section in the introduction to this topic in the Teacher’s Resource Book (TRB)
• Mental Maths handbook for Multiplication and Division from the PDST
• Number Talks book by Sherry Parrish
• Splat! Similar to Number Talks, these free resources from Steve Wyborney encourage discussion and reasoning. Play the PowerPoint presentations on your class IWB while the children use their Operation Maths MWBs to respond.
• Multiplication and Division Board on Pinterest
• Does the decimal point move? A one minute video which shows multiplying/dividing by 10/100 etc using the moving digits approach
• Can I divide by zero? A one minute video which illustrates the difficulties that arise when we consider dividing by zero
• Two more short videos from Graham Fletcher showing the progression of multiplication and division.