# Dear Family, your Operation Maths guide to Money

## Dear Family, your Operation Maths guide to Money

Category : Uncategorized

Dear Family, listed below are some practical suggestions as to how you might support your children’s understanding of the maths topic of money. Also below, are a series of links to digital resources that will help both the children, and you, learn more about money. The digital resources are organised according to approximate class level.

#### Practical Suggestions for all Children

• Difficult Topic? Although money plays a very common, and perhaps very important, part in every child’s life, money is not automatically easy or obvious to learn about:
• It comes in different colours, shapes and sizes, and in metal and paper forms (i.e. coins and notes), each of which has its own value.
• The sizes of the coins and notes are NOT proportional to their value i.e. a 20c coin is not twice as big as a 10c coin; a €100 note is not ten times the size of the €10 note.
• Money can be expressed using the symbols € or c, but NOT using both at the same time. Sometimes there’s a decimal point; sometimes there’s not. And, when using the € sign, it comes first (even though €6 is said as “six euro” as opposed to “euro six”), whereas the c sign comes after the numeral.
• So even though understanding and using money is a vital life skill, it can’t be taken for granted that children will easily “get” this understanding.
• Cash starved: More and more, transactions are becoming cashless, as people use credit/debit cards, money apps, contactless and online payments more than ever before. In the recent past, coins and notes, were very much a regular part of a child’s experiences; watching others counting out coins and notes to pay for goods, perhaps handing over a larger amount than required and watching change being handed back. Because of cashless transactions, today’s children are missing out on essential opportunities to handle cash, and/or see it being handled in real-life situations. The increased use of plastic and contactless payments also limits the opportunities for people to use their maths skills to total mentally, calculate change etc.
• Cash is King: Since today’s children have less exposure to cash transactions, where possible, allow your child to handle real coins and notes.
• Collect coins in a jar and invite your child to count the money every so often to find out the total. Are there different ways to count mixed coins? What strategies might be better (more efficient)?
• If your child is not able to count up the total of amounts yet, then ask them to sort the coins into groups e.g. brown coins, gold coins, coins with two colours, coins with 1 on them, with 2 on them, with 5 on them etc (see also the Coins Game below). In particular, draw their attention to the fact that all the euro coins and notes only start with the digits 1, 2 or 5 and may or may not be followed by one or two zeros.
• Play shop at home. Use empty food containers etc., as goods to be bought/sold. Use play money or real coins for cash.
• If shopping with cash, involve your child: get them to handle the money, to identify the coins and/or most suitable amount to hand to the cashier and to predict (roughly or accurately, depending on the ability of the child) the change due back.
• If your child gets pocket money encourage them to talk about how much they have, how much they spend, how much they have left.
• Can you afford it? Should you buy it? While cashless transactions might be more regular than cash ones, one thing that has remained constant is that we all still appreciate the value of money, and getting value for our money.
• Encourage your child to budget and save for upcoming events (Christmas, holidays etc.) and/or to purchase more expensive items ( eg bike, games console, phone etc). Discuss the amount of money required, how much they currently have, how much they could expect to earn and/or save each week, how long it will take for them to have the necessary amount. Encourage them to write down this information as a type of written budget or financial plan (see also the Budget Game below, for 3rd class up).
• When purchasing items encourage your child to consider its value, its cost, and whether a similar item be purchased elsewhere for less. Shop around, as they say, to research your options, whether in the actual or virtual (online) shops.
• When grocery shopping, keep an eye out for the advertised special offers and deals; are they good options? What about multi-packs; are they good value? If there are different multi-pack offers for the same product, which offer is the best value? But don’t forget that just because there is good value on offer, if we end up buying more of the product than we need, will it end up going to waste?
• Go through the till receipt together after shopping; what did you buy? What items cost the most? What items cost the least? What products cost about the same amount? Was there any product that you hadn’t realised cost so much or so little?
• When out shopping for clothes, give your child a limit to the amount that can be spent. It is amazing how value-driven this can make your child become, and more selective of what they will purchase!
• If you are comfortable allowing your child to use the internet, he/she could help research a holiday or break for the family. Examine together which destination has the best deal/offers, etc.
• Money makes the world go round! Children may not realise that, outside of the Euro Zone, most countries have their own currency. And they may also not realise that, when changing currency, you cannot do a straight swap i.e. €1 doesn’t equal £1 or \$1; the new value must be calculated using an exchange rate, which also varies. If going on holiday to a non-Euro Zone country, involve your child in researching the exchange rate, and calculating how much of the foreign currency they will get when they exchange their euro.

#### Digital Resources for Infants

Spot the coins: Beginner level: Find the coins hidden in each picture. Advanced level: find the coins and order them according to value.

Coins game: Click on the Euro flag to select euro coins. Start with Sorting to sort One Coin or Two Coins into the money box(es). Next try Ordering and Counting money. Start with the easier options in each section and move on if too simple and/or when confident.

Toy Shop: Work out which coins will buy toy shop items, using just One Coin or Mixed Coins. In the Mixed Coins option you can also calculate change.  Start with the easier options and move on if too simple and/or when confident.

Keep Helen’s money safe: Read the story and decide what Helen should do with her money to keep it safe. Play ages 5-6

Moneyville is a fun and entertaining online virtual world that gives your child a basic understanding of the value of money and the basic principles behind earning and spending money. Suitable for children of 5 years and up.

My Money Week: Run every year in the UK around May, this is a national activity week which aims to boost children’s skills, knowledge and confidence in money matters. To access the resources, you need to set up a free account, which requires email details etc and entering any UK postcode. Once registered and logged in, scroll down to the bottom of the primary resources and click on Start journey; this will start off a series of excellent videos on Max’s Day Out, in which Max is deciding how best he might spend the money that he got for his birthday. The videos are designed in such a way that each one presents two possible options; the viewer selects an option, which automatically brings them to the follow-up video for their choice. There are many other resources also available here that focus on managing money.

Values, Money and Me: This is a set of stories, activities and quizzes that encourage children to consider what money can and cannot buy. These stories are particularly suited to younger children: Finders Keepers, I want it, Do the Right Thing, Charity Job Week & Hero or Zero.

Money Match: Match the coins to the correct value. As some of these coins may be less familiar to children in the infant classes, encourage them to match the easier coins first. (Not tablet friendly – requires Adobe Flash Player).

Money Match Amounts: Match the coins to the correct amount. (Not tablet friendly – requires Adobe Flash Player).

Count up: Count up the value of the coins shown. Use the + and – buttons to enter the value and then check. For infants, change the option on the bottom left side of the screen to use coins up to and including 2c initially, and then increase the options as the child gets more competent. (Not tablet friendly – requires Adobe Flash Player).

Coin Exchange: Swap the coin shown for the stated number of coins of lesser value. For infant classes, change the option on the top left side of the screen to exchange coins up to and including 10c initially, and then increase the options as the child gets more competent. (Not tablet friendly – requires Adobe Flash Player).

Money: 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

Spot the coins: Beginner level: Find the coins hidden in each picture. Advanced level: find the coins and order them according to value.

Coins game: Click on the Euro flag so select euro coins. Start with Sorting to sort One Coin or Two Coins into the money box(es). Next try Ordering and Counting money. Start with the easier options in each section and move on if too simple and/or when confident.

Toy Shop: Work out which coins will buy toy shop items, using just One Coin or Mixed Coins. In the Mixed Coins option you can also calculate change.  Start with the easier options and move on if too simple and/or when confident.

Moneyville is a fun and entertaining online virtual world that gives your child a basic understanding of the value of money and the basic principles behind earning and spending money. Suitable for children of 5 years and up.

Keep Helen’s money safe: Read the story, decide what Helen should do with her money to keep it safe, and keep a record of the money that she gets along the way. Play ages 7-8.

My Money Week: Run every year in the UK around May, this is a national activity week which aims to boost children’s skills, knowledge and confidence in money matters. To access the resources, you need to set up a free account, which requires email details etc and entering any UK postcode. Once registered and logged in, scroll down to the bottom of the primary resources and click on Start journey; this will start off a series of excellent videos on Max’s Day Out, in which Max is deciding how best he might spend the money that he got for his birthday. The videos are designed in such a way that each one presents two possible options; the viewer selects an option, which automatically brings them to the follow-up video for their choice. There are many other resources also available here that focus on managing money.

Values, Money and Me: This is a set of stories, activities and quizzes that encourage children to consider what money can and cannot buy. These stories are particularly suited to younger children: Finders Keepers, I want it, Do the Right Thing, Charity Job Week & Hero or Zero.

Coin Cruncher: In this game you either select the correct coins to Make the Total or select the correct value for How much? There is an Easy level (no timer) and a Hard level (same question types but with a timer).

The Change Game: Click on the correct amount of change that you should get back.

That Quiz – Money: This quiz has lots of options, on the left hand side, that can be changed to suit the ability of the child. Ensure that the currency is set to Euro and the level is set to 1. 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 three different types of activities: For Identify (it automatically starts on this) you must type in the value of the cash shown; if you set it to Compare you must click on the amount of greater value; if you set it to Make change you must click on the cash required to make the correct change for the given transaction.

Adding Money Values: This video from Operation Maths allows the children to practice their addition of money skills.

Custom Car Garage: Select and pay for car accessories, using the correct coins. For first and second class, start at level one initially, and then go up levels as the child gets more competent.

Money Match: Match the coins to the correct value. (Not tablet friendly – requires Adobe Flash Player).

Money Match Amounts: Match the coins to the correct amount. (Not tablet friendly – requires Adobe Flash Player).

Count up: Count up the value of the coins shown. Use the + and – buttons to enter the value and then check. For first and second class, change the option on the bottom left side of the screen to use coins up to and including 10c initially, and then increase the options as the child gets more competent. (Not tablet friendly – requires Adobe Flash Player).

Shopping Spree: Choose the correct coins to purchase the presents. For first and second class, change the option on the bottom left side of the screen to buy items costing between 10c- 99c initially, and then increase the options as the child gets more competent. (Not tablet friendly – requires Adobe Flash Player). You can also play a timed version of this game here.

Change Please: Choose the correct coins to make the change required. (Not tablet friendly – requires Adobe Flash Player). You can also play a timed version of this game here.

Coin Exchange: Swap the coin shown for the stated number of coins of lesser value. Change the option on the top left side of the screen to exchange coins up to and including 10c or 20c initially, and then increase the options if too simple or as the child gets more competent. (Not tablet friendly – requires Adobe Flash Player).

Coconut Ordering Game: Select Prices and € to order amounts of euro

Money: 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 to Sixth Classes

Money: Background information on money from Maths is Fun, including currencies, finding unit price, interest, investing money, etc. Often there are also related activities.

My Money Week: Run every year in the UK around May, this is a national activity week which aims to boost children’s skills, knowledge and confidence in money matters. To access the resources, you need to set up a free account, which requires email details etc and entering any UK postcode. Once registered and logged in, scroll down to the bottom of the primary resources and click on Start journey; this will start off a series of excellent videos on Max’s Day Out, in which Max is deciding how best he might spend the money that he got for his birthday. The videos are designed in such a way that each one presents two possible options; the viewer selects an option, which automatically brings them to the follow-up video for their choice. There are many other resources also available here that focus on managing money.

Values, Money and Me: This is a set of stories, activities and quizzes that encourage children to consider what money can and cannot buy. These stories are particularly suited to older children: Finders Keepers, I want it, Charity Job Week, a Fair Day’s Pay, Do You Need It? & Costing the Earth.

Is it a bargain? This fun mini-maths lesson gets pupils to use their mathematical ability to work out if so called ‘special offers’ are in fact good deals.

That Quiz – Money: This quiz has lots of options, on the left hand side, that can be changed to suit the ability of the child. Ensure that the currency is set to Euro and the level is set to 1. 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 three different types of activities: For Identify (it automatically starts on this) you must type in the value of the cash shown; if you set it to Compare you must click on the amount of greater value; if you set it to Make change you must click on the cash required to make the correct change for the given transaction.

Coconut Ordering Game: Select Prices and € to order amounts of euro

Space Trader: Practise spotting value for money as you trade with three outlandish alien shopkeepers for a range of space commodities.

Count up: Count up the value of the coins shown. Use the + and – buttons to enter the value and then check. Change the option on the bottom left side of the screen to use coins up to and including 50c initially, and then increase the options if this is too simple or as the child gets more competent. (Not tablet friendly – requires Adobe Flash Player).

Shopping Spree: Choose the correct coins to purchase the presents. Change the option on the bottom left side of the screen to buy items costing between €1.00 – €5.00, and then increase the options if too simple or as the child gets more competent. (Not tablet friendly – requires Adobe Flash Player). You can also play a timed version of this game here.

Change Please: Choose the correct coins to make the change required. (Not tablet friendly – requires Adobe Flash Player). You can also play a timed version of this game here.

Coin Exchange: Swap the coin shown for the stated number of coins of lesser value. Change the option on the top left side of the screen to exchange coins up to and including 10c or 20c initially, and then increase the options if too simple or as the child gets more competent. (Not tablet friendly – requires Adobe Flash Player).

Calculating Unit Rates: For 5th & 6th classes, this visual video from MashUp Math explains how to calculate unit rates, which has applications in unitary value in money, averages, ratios and proportion etc

Find the Total Cost: Again for 5th & 6th classes, this video from MashUp Math explains how to find the total cost, involving sales tax, ratios and proportions.

The Budget Game: Older children can explore the realities of budgeting, income and expenditure as well as how their own choices affect their money, well-being and enjoyment balances.

Money: 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 … Capacity (all classes)

Category : Uncategorized

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

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

### Initial exploration – CPA approach

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

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

From Operation Maths 1

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

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

From Operation Maths 1

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

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

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

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

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

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

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

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

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

### Using more accurate measures

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

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

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

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

### Renaming units of capacity

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

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

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

### Capacity & Volume

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

From Operation Maths 6

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

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

From Operation Maths 6

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

Category : Uncategorized

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

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

### Initial exploration – CPA approach

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

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

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

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

PS: The video title says it’s a homemade scale but, strictly speaking, this is not a scale as a scale allows you to measure and record a specific measure.

HINT: constructing one of these could make for a great alternative homework task, with added benefits! Let’s face it: what child will construct one without wanting to try it out there and then? Doing maths without even being asked!

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

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

### Using scales: estimating & measuring

From Operation Maths 5, Pupils’ Book

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

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

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

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

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

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

### Renaming units of weight

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

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

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

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

Category : Uncategorized

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

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

In Operation Maths:

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

### Measuring area

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

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

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

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

### Estimation and efficiency

When using both non-standard and, later, standard square units, the children should always be encouraged to estimate the area first before measuring. As mentioned previously in the post on Length, rather than estimating the area of A, B, C and D before measuring A, B, C and D, it would be better if the children estimated the area of A and then measured/counted the area of A, estimated the area of B and then measured/counted the area of B and so on. Thus, they can reflect on the reasonableness of their original estimate each time and use this to refine their next estimate so that it might be more accurate. In this way, the children will also begin to develop their sense of space.

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

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

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

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

### Calculating area

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

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

### Area and perimeter

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

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

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

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

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

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

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

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

## Digging Deeper into … Length (all classes)

Category : Uncategorized

### Initial exploration of Length

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

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

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

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

### Non-standard and standard units

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

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

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

### Estimating & measuring

The children should always be encouraged to estimate before measuring. And rather than estimating the length of A, B, C and D before measuring A, B, C and D, it would be better if the children estimated the length of A and then measured the length of A,  estimated the length of B and then measured the length of B and so on. Thus, they can reflect on the reasonableness of their original estimate each time and use this to refine their next estimate so that it might be more accurate.  As mentioned previously, the child’s ability to connect standard units of length to themselves, not only helps them internalise a sense of length, but to also use this sense to produce more accurate estimates.

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

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

### Renaming units of length

From third class on, the children will be expected to rename units of length appropriate to their class level. While changing 136 cm to 1m 36cm or 1.36m will typically be done correctly, converting figures which require zero as a placeholder (eg 1m 5cm, 2.4m ) are quite often problematic and can reveal an underlying gap in understanding, that is not revealed by the more obvious measures. In these cases, the children should be encouraged to return to the concrete and pictorial experiences as a way of checking the reasonableness of their answers, eg:

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

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

### Perimeter and area

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

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

### Scale drawings

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

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

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

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

## Digging Deeper into … Money (all classes)

Category : Uncategorized

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

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

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

### Elicit prior knowledge & concrete exploration

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

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

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

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

### Exploring the value of the coins

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

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

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

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

### Mental calculations with money

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

### Visual Strategies for Problem Solving

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

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

### Other tips and suggestions for teaching money

• Emphasise that money is based on the euro. Cent coins are merely fractions of that unit; euro coins and notes are multiples of that unit. In this way money can also be used to help teach other strand units, including place value, operations and decimals (see examples below).
• Using money to represent place value with whole numbers and decimals

Using money to model 2-digit division

Using money to model decimals to hundredths

• Emphasise the importance of using an efficient estimation strategy when calculating with money:
• front-end estimation: where you only consider the most significant digit i.e. think of €26.95 as €20
• rounding: where you round to the nearest unit, ten etc, i.e. think of €26.95 as €27 or as €30. Rounding produces more accurate estimates than front-end estimation; however, it can also be more time-consuming for some children and thus less efficient.
• Value for money: Encourage the children to compare prices in different shops and/or catalogues to identify the best price and when comparing items being sold as multiples to compare their values using the unitary method and T-charts, as shown above.
• Use That Quiz for an interactive quiz based on euro. Use the options on the left hand side to change the difficulty (level) and the scope of questions (identify, compare or make change). For more on using That Quiz in the classroom read on here.

## Digging Deeper into … Time (all classes)

Category : Uncategorized

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

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

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

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

### Telling Time

Telling time requires the children to:

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

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

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

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

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

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

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

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

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

### Time as Duration

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

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

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

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

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

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

### Other tips and suggestions for teaching time

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