Author Archives: Operation Maths

Maths by Month – June

Category : Uncategorized

The summer holidays are in sight!

In this June overview for Operation Maths users, there are links to topic-specific posts and articles, as well as a whole host of extra suggestions, links etc. To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the blog via email, on the top right hand of this page.

Pssst! Book lists not finalised yet? Please consider Operation MathsNumber Facts, Bua na Cainte & Exploring Spelling. Click on the links for more information and to view sample pages from each program and/or contact your local Edco reps for samples.

Operation Maths Jr Infs to 2nd classes:

  • Junior Infants will be reinforcing their understanding of the numbers 0-5 via the topic of money
  • Senior Infants will continue to consolidate their understanding of the numbers to 10, via combining and partitioning activities (including the use of the Operation Maths ten frames), solving problems using the number line and representing and interpreting data in a block graph. The children will be exploring patterns to discover different arrays of the same number, patterns with colour and numbers and odd and even numbers. They are also learning to read time in one-hour intervals.
  • First Classes will be exploring Weight , re-visiting Patterns as well as exploring 3-D objects and, in particular, will be connecting their understanding of 3-D objects to their understanding of 2-D shapes.
  • Second Classes will be extending their ability to read, write and order numerals to 199, whereby they will also be exploring their understanding of place value. They will also be introduced to Area for the first time. Towards the end of the month, as part of spatial awareness, they will revisit the half turns and quarter turns that they originally met as part of Lines and Angles 

(click on any of the links above for more information)

Operation Maths users can also access a class specific, month-by-month list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 

  1. Log into your edcolearning account
  2. Click on the At School Book for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select Weblinks from list of categories and then click to download the document.

Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above.

 

 

Operation Maths 3rd to 6th classes:

Operation Maths 3-6 is specifically structured so that the programme can be completed by the end of May, thus covering all of the topics in advance of the standardised testing.

So you might find yourself looking for inspiration to fill the maths lessons from now until the end of month. Whether you’re an Operation Maths user or not, there are a whole suite of suitable ideas on this blog post.

Are you an Operation Maths user? What do you like about the programme? What suggestion do have for how it might be improved? We’d love to hear your views! Please leave a comment on this Facebook post


Maths for June

Category : Uncategorized

Hooray! June is here! You can almost smell the summer holidays!

And soon the testing will be all over (if it’s not already) and the books will be finished (if they’re not already). If you’re a user of Operation Maths 3-6 you are quite likely to be finished your books, as the programme is designed to be completed by the end of May, so as to have it all covered in advance of the standardised testing.

So now you might find yourself looking for inspiration to fill the maths lessons from now until the end of month. Whether you’re an Operation Maths user or not, look no further than the following ideas.

For Operation Maths users:

If you hadn’t had a chance to dip into these specific features of the Operation Maths programme so far this year then why not try these out now?

  • Let’s Investigate! These sections are the last one or two pages at the end of the Pupils’ Books ( for third to sixth classes) where the focus is on open-ended problems. Some of these are “big” enough to fill a whole lesson, others might become additions to a lesson or be combined to become a lesson. The children could also select which particular investigation(s) they’d like to explore either a whole class or with individual groups selecting different investigations, with results to to be communicated back to whole class when complete.
  • Early Finishers Photocopiables: These can be found in your Teachers Resource Book (TRB) and can also be  a great way to help deepen the children’s understanding of a topic covered earlier in the year. For 3rd to 6th classes, problem solving is also an integral part of these activities. In the TRBs for Junior Infants to 2nd classes, there are both Early Finishers photocopiables and dedicated problem-solving activities.
  • Maths Around Us: If your class has access to recording devices, why not challenge them to make their own Maths Around Us video based on maths content they covered this year. Watch some of the Operation Maths Maths Around Us videos on www.edcolearning.ie for inspiration.
  • As mentioned in a previous post, don’t feel under pressure to complete all of the above activities, only just what appeals most to you or is most suited to your class.

For everybody!

  • Change their attitude to maths generally: Most people have this belief that there is such a thing as a maths brain, a belief which Jo Boaler, among others, strongly challenges. In conjunction with her youcubed team at Stanford University, in 2015 they put together resources, videos etc for a Week of Inspirational Maths and followed that up with a Week of Inspirational Maths 2 in 2016, the latter of which has lessons and activities aimed at infants to 6th, as well as second level. Click on the link for an overview of the activities in Week of Inspirational Math 2, which also includes links to all the required resources.
  • Take time to problem-solve: often, during the school year, time is at a premium, yet Dan Finkel argues in this TEDx Talk that “allowing children time to struggle” is one of the Five Principles of Extraordinary Math Teaching. So after watching this video, why not present the images he uses to a 5th or 6th class and give them time to “notice and wonder”. The children could use sentence/questions stems like “I notice that…” and ” I wonder why/how/what ….” to get them thinking and discussing. Read on here for more sources of deep and rich problems.
  • Try out a new methodology with your class. It can be a good idea to try out something new in June when there’s less pressure to succeed and you’re familiar with your class, rather than trying out something new in September when you’re trying to get to grips with new class, new books, perhaps new room etc!  One initiative I would wholeheartedly recommend is Number Talks. You could do a number talk with your class aimed at their current level or challenge them to do a number talks session aimed at the class they’ll be in next September.
  • Do a maths project. In the Maths Curriculum Teacher Guidelines (DES, 1999) maths projects are listed as one of the examples of maths problems that we are encouraged to incorporate into our teaching. It can be difficult to include maths projects earlier in the year when the pressure is on to cover the content, making June an ideal time to explore them. For 10 “awesome” ideas, check out this post from the Mashup Math blog. I particularly like the idea of the child planning out their ideal holiday; so much real-life maths, costs, budgeting, estimating costs of luggage, time needed to get to the airport, distance from destination to airport etc. The NCETM Primary and Early Years Magazine also has suggestions for projects, the first one again focusing on financial education.
  • Take it outdoors. Another type of maths problem listed in the Teacher Guidelines is maths trails. If the rain stays away for long enough why not get outside and do some maths trails? Or if you teach a more senior class, why not get them to design a maths trail for a junior class based on the school grounds or nearby environment. For more trail ideas read on here.
  • Maths is Magic! There is a lot of mathematics behind magic. You could give the children magic tricks to investigate. Check out this article, again from the NCETM Primary and Early Years Magazine for sites to explore.
  • Break the code: Explore the maths behind codes and code-breaking. You could ask the children to make up their own codes and crack a friend’s. Click here for links to suitable sites.
  • Have a maths game-themed day. Another one of Dan Finkel’s Five Principles of Extraordinary Math Teaching is play. Most games and puzzles are mathematical in nature. Get the children to bring in a favourite game from home, to play in class, that requires mathematical thinking. Alternatively, get them to research a suitable one on the internet.

Maths by Month – May

Category : Uncategorized

In this May overview for Operation Maths users, there are links to topic-specific posts and articles, as well as a whole host of extra suggestions, links etc. To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the blog via email, on the top right hand of this page.

Pssst! Thinking about book lists? Please consider Operation MathsNumber Facts, Bua na Cainte & Exploring Spelling. Click on the links for more information and to view sample pages from each program and/or contact your local Edco reps for samples.

Operation Maths Jr Infs to 2nd classes:

  • Junior Infants will be exploring Weight and Capacity as part of measures. They will also continue to consolidate their understanding of numeration,  counting and combining within 5, looking specifically at the numeral zero, combining sets of numbers to 5 and identifying related facts (often referred to as turn-around facts in the higher classes). As mentioned previously, the  Number Talks resources  for the numbers to 5 available at the link are very applicable as well as the practical activities and stations using concrete materials, that are suggested in the TRB.
  • Senior Infants will continue to consolidate their understanding of the numbers to 10, via counting and numeration, comparing and ordering, and the combining and partitioning of sets using a variety of models including the Operation Maths ten frames. As mentioned previously, the Number Talks resources for the numbers to 10 available at the link are very applicable as well as the practical activities and stations using concrete materials, that are suggested in the TRB. Extra resources that could also be used to consolidate these concepts include the Splat! resources made freely available here by Steve Wyborney. This link will bring you to his Splat! resources for the numbers 3 to 10, which are ideal for senior infants at this stage of the year. Play the PowerPoint presentations on your class IWB while the children use their Operation Maths MWBs to respond.

Senior Infants will spend the second half of the month looking at Money.

  • First Classes will be exploring deeper into Money and 2D shapes. Towards the end of the month, the children will be exploring operations again, but this time to include addition with renaming. As mentioned previously (eg in Maths by Month in March), on many of the pages in the At School books the calculations are presented horizontally. This is deliberately done to encourage the children to complete the calculations using concrete materials, pictorial representations and/or mental strategies, as opposed to always using the vertical column method. While the development of traditional written procedures (eg the column method) is still important, these written methods are not more important than the development of mental computation skills and the ability to visualise and manipulate numbers mentally.

Some of the mental strategies for addition and subtraction, used in Number Talks, would also be every useful here e.g. partitioning, friendly numbers, making tens (compensation) and removal (deduction). Suitable number strings can be accessed at the link above and of course teachers can make their own number strings based on the horizontal calculations in their At School book.

  • Second Classes will exploring Operations (addition & subtraction within 99, without and with renaming), Money and Place Value to 199.  
  • Regarding operations, and as mentioned in Maths by Month in March, on many of the pages in the At School books the calculations are presented horizontally. This is deliberately done to encourage the children to complete the calculations using concrete materials, pictorial representations and/or mental strategies, as opposed to always using the vertical column method. While the development of traditional written procedures (eg the column method) is still important, these written methods are not more important than the development of mental computation skills and the ability to visualise and manipulate numbers mentally.Some of the mental strategies for addition and subtraction, used in Number Talks, would also be every useful here e.g. partitioning, friendly numbers, making tens (compensation) and removal (deduction). Suitable number strings can be accessed at the link above and of course teachers can make their own number strings based on the horizontal calculations in their At School book.

(click on any of the links above for more information)

Operation Maths users can also access a class specific, month-by-month list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 

  1. Log into your edcolearning account
  2. Click on the At School Book for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select Weblinks from list of categories and then click to download the document.

Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above.

 

 

Operation Maths 3rd to 6th classes:

The topics for this month are:

Click on each link above to access more in-depth information and links on each of the topics for this month.

May is also the time when teachers start thinking about the upcoming standardised tests and about doing some revision. You could read this post for some tips and suggestions for the same.

Operation Maths users can also access a class specific, topic-by-topic list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 

  1. Log into your edcolearning account
  2. Click on the Pupil’s Book for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select Weblinks from list of categories and then click to download the document.

Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above. 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.


Digging Deeper into … Number Sentences, Equations & Variables (3rd – 6th)

Category : Uncategorized

In the Primary Mathematics Curriculum (1999), this topic appears as three separate strand units, all within the strand of Algebra:

  • Number Sentences (3rd & 4th class)
  • Equations (5th & 6th class)
  • Variables (6th class)

However, since these concepts are intrinsically connected, in Operation Maths they are taught in a cohesive and progressive way through third to sixth class.

  • Number sentences are mathematical sentences written using numerals (e.g. 1, 5, 67, 809, 1.45, 1/2  etc.) and mathematical symbols (e.g. +, -, x, ÷, <, >, =).
    • They include both equations (see below) and inequalities (64 < 82, 23 > -16), although the term inequalities is not specifically used.
    • The unknown or missing value in a number sentence (i.e. a variable) can be represented by a frame (box), by a shape, or by a letter, although it should be noted that the Primary Mathematics Curriculum (1999) specifies a preference for a frame (box), up to the introduction of variables in 6th class
  • An Equation is a special type of number sentence, containing an equals sign, to show that two expressions are equal (e.g. 5 = 3 + 2, 5 + 6 = 20 – 9, etc.)
  • A variable is a value in an expression that can  change or vary. However, when there is only one variable in an equation then the value of that variable can be calculated e.g. a + 6 = 9, 20 = 4b.

Thus, while these strand units are only being formally introduced from third class on, the children have actually been exposed to number sentences, equations and variables (i.e. the frame) since the infant classes.

 

Equations

(aka Number sentences with an equals sign)

Understanding equations necessitate the appreciation of the correct meaning of the equals symbol. Many children incorrectly translate the equals symbol (=) as meaning ‘and the answer is…’. This incorrectly reinforces that both its purpose and position is to precede the answer in any calculation, a misconception also reinforced by calculators, where you press the = button to get the answer. Such misunderstanding is
evident when you see responses like these:
5 + 6 = [11] + 3 , i.e. ‘5 + 6 is 11’
5 + 6 = [14] + 3 , i.e. ‘5 + 6 + 3 is 14’
Adults may also unwittingly compound this, by using ‘makes’ or ‘gives’ as a synonym for equals.

It is vital that the children recognise that the equals symbol indicates that both sides of the equation (which will be referred to simply as a number sentence until fifth class, when the term “equation” is introduced), are equal to one another/are the same value/are balanced. In this way an actual balance (pan or bucket) and cubes can be extremely valuable to model (and solve) equations e.g. in the images below, the first balance shows that 5 equals a group of 3 and a group of 2, and the second balance shows that 12 equals 3 groups of 4.

From Operation Maths 4

From Operation Maths 5

Furthermore, teachers should reinforce the correct meaning for the symbol = by only translating it as ‘equals’, ‘is equal to’ and/or ‘is the same as’.

 

Inequalities

(aka Number sentences with greater than/less than sign)

Despite the fact that the children have been using the greater than and less than symbols since 2nd Class, many still have difficulties reading them and interpreting their meaning. Using a balance and concrete materials, in a similar way as when teaching equations, can greatly help children to gain deeper understanding of the symbols and their meanings.

From Operation Maths 4

Through exploration they can identify what is the maximum number of cubes they can put on a side that is less than the other side, before it makes the balance tip in the other direction, thereby invalidating the number sentence; or the minimum number of cubes they can put on a side that is greater than the opposite side, so as to keep the number sentence true.

 

Using estimation strategies

Often, when having to indicate if a given number sentence is true or false, it is not always necessary for the children to calculate both sides of the number sentence exactly. There is (usually) only one true or correct option, meaning that every other answer is incorrect or false. Encourage the children to use their estimation and number sense skills to quickly recognise when a statement is obviously false, e.g. a big difference in the size of numbers on one side versus the other.

While some might view this as a type of ‘cheat’ strategy, in truth, it is more about identifying the most efficient approach, while also reinforcing the value of estimation in general and, particularly, as a way to make calculations easier.

 

Translating number sentences into word problems and vice-versa

As mentioned earlier, this is in fact a skill that the children would have been exposed to, and been using, since the infant classes. Furthermore, as this topic is deliberately positioned towards the end of the yearly plans in Operation Maths 3-6, the children will have already been using this skill very regularly in the number, data and measures chapters, prior to this point of the school year.

The curriculum specifies that the children should be enabled to translate number sentences into word problems, both of which can be viewed as abstract representations. Worth noting, is that the curriculum doesn’t emphasise the importance of the translating the number sentences and word problems into concrete and/or pictorial representations. Whereas, in Operation Maths, (in keeping with its overarching CPA approach) , there is significant emphasis placed also on utilising various concrete materials and visual strategies to represent the word problems and number sentences.

From Operation Maths 5

The development of visual strategies for problem-solving,  is a central focus of the work throughout the Number chapters. Thus, this topic allows the teacher to revise the visual strategies covered so far and assess how well the children understand them and can apply them.

The interconnectedness of real-life scenarios and mathematical sentences/equations should also be emphasised. At primary level, there should always be some relatable context for any number sentence.
For many children, when looking at a number sentence, it can be difficult to appreciate how a collection of digits and symbols could relate to a real-life scenario with which they can identify. That is essentially what a word problem is; it provides a real-life context within which to frame the numbers and operators involved. Emphasise to the children throughout this topic how the number sentences could be given a real-life story (i.e. word problem), and encourage them to come up with possible stories either verbally or written down.

And, depending on the context given to a particular story, the visual representation may also be different, even though the number sentence/equation may stay the same. For example for the number sentence 7 – 4 = ? the word problem (context) could be either of the following:

  • Ali had 7 cookies. He ate 4 cookies. How many cookies does he have left?
  • Áine has 7 cookies. Abdul has 4 cookies. How many more cookies has Áine than Abdul?

Image created using Bar Modelling eManipulative, accessible on edcolearning.ie

And while the number sentences are the same, both the contexts and the pictorial representations (e.g. bar models, as shown above) are different, as they represent different types of subtraction. In the case of the first word problem, this is describing subtraction as deduction, and a part-whole bar model is more suitable. In the case of the second word problem, this is describing subtraction as difference, and a comparison bar model is more suitable.

 

Identifying operation phrases

When the children are translating word problems into number sentences, it is very important that they can understand the context being described and are able to identify that phrases that indicate the operation(s) required.

Regularly interspersed throughout the operations chapters in the Discovery books for Operation Maths 3-6,  there are activities which enable the children to identify and colour-code the specific vocabulary that an indicate the required operation (see example below). This topic provides an ideal opportunity to review this skill and assess/re-teach the children accordingly.

From Operation Maths 4, Discovery Book

In particular, many of the Talk Time activities, require the children to suggest ways to verbalise the various equations, e.g:

  • The difference between 46 and 18 is equal to the product of 4 and 7; true or false?
  • 18 subtracted from 46 equals 4 times 7; true or false?

Where possible the children should suggest alternative phrases for the same equation thus reinforcing the use of correct mathematical language.

 

Input and Outputs

In Operation Maths 4 & 5, activities based on inputs and outputs are included as a means to consolidate the children’s understanding of number sentences and their ability to write number sentences (see below).  Input-output activities can provide great scope for problem solving, as well as preparing the children for calculations involving variables in sixth class.

From Operation Maths 4

 

Variables

Variables are formally introduced in sixth class, although the children have encountered variables (as a symbol or shape to represent a missing value) since they first encountered the frame (answer box).

When calculating with variables, both part-whole bar models and comparison bar models can be very useful to represent the relationship between the known and unknown values.

From Operation Maths 6

 

 


Digging Deeper into … Capacity (all classes)

Category : Uncategorized

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

Further Reading & Viewing:


Maths by Month – April

Category : Uncategorized

In this April overview for Operation Maths users, there are links to topic-specific posts and articles, as well as a whole host of extra suggestions, links etc. To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the blog via email, on the top right hand of this page.

Pssst! There is still time to register for the April Edco Primary Publications launches for 2018 , which will be taking place in Kilkenny (rescheduled because of the snow), Galway, Athlone and Wexford. As well as launching Number Facts 1-4 and Bua na Cainte 3, they will also be showcasing Operation Maths and Exploring Spelling. Click on the link above for more info and to register. If you want to find out more about the Number Fact series you can read this comprehensive and independent review from the Irish Primary Teacher blog and/or you can check out the Number Facts site.

Operation Maths Jr Infs to 2nd classes:

  • Junior Infants will be exploring ordinal number (first, last), comparing and ordering numbers 1-5, as well as combining sets of numbers to 5. As usual, a whole suite of supporting activities are detailed in the TRB as well as ideas for Aistear play.
  • Senior Infants will be consolidating their understanding of the numbers to 10, via counting and numeration and the combining and partitioning of sets using a variety of models including the Operation Maths ten frames. As mentioned previously, the  Number Talks resources  for the numbers to 10 available at the link are very applicable as well as the practical activities and stations using concrete materials, that are suggested in the TRB. Extra resources that could also be used to consolidate these concepts include the Splat! resources made freely available here by Steve Wyborney. This link will bring you to his Splat! resources for the numbers 3 to 10, which are ideal for senior infants at this stage of the year. Play the PowerPoint presentations on your class IWB while the children use their Operation Maths MWBs to respond.

Senior Infants will also be exploring the strand unit of capacity toward the end of the month.

  • First Classes will be looking at Time and 2D shapes. This work will largely centre around half and halves, (eg time to half hours and partitioning shapes into halves) as introduced formally in the strand unit of Fractions in March

Towards the end of the month, in Operations, the children will be exploring addition and subtraction without renaming. This is an extension of the activities completed in March as the operations are now up to the first class limit of 99.

As mentioned in Maths by Month in March, on many of the pages in the At School books the calculations are presented horizontally. This is deliberately done to encourage the children to complete the calculations using concrete materials, pictorial representations and/or mental strategies, as opposed to always using the vertical column method. While the development of traditional written procedures (eg the column method) is still important, these written methods are not more important than the development of mental computation skills and the ability to visualise and manipulate numbers mentally.

Some of the mental strategies for addition and subtraction, used in Number Talks, would also be every useful here e.g. partitioning, friendly numbers, making tens (compensation) and removal (deduction). Suitable number strings can be accessed at the link above and of course teachers can make their own number strings based on the horizontal calculations in their At School book, for example, from page 93 (four calculations in each number string works well) : 39 – 7, 39 – 0, 39 – 1, 39 – 9

  • Second Classes will exploring 2D shapes, Capacity and 3D shapes and angles. Angles are introduced for the first time in 2nd class, and the concept is linked closely to the children’s understanding of fractions, spatial awareness and 2D shapes.

(click on any of the links above for more information)

Operation Maths users can also access a class specific, month-by-month list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 

  1. Log into your edcolearning account
  2. Click on the At School Book for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select Weblinks from list of categories and then click to download the document.

Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above.

 

 

Operation Maths 3rd to 6th classes:

The topics for this month are:

Click on each link above to access more in-depth information and links on each of the topics for this month.

Operation Maths users can also access a class specific, topic-by-topic list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 

  1. Log into your edcolearning account
  2. Click on the Pupil’s Book for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select Weblinks from list of categories and then click to download the document.

Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above. 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.


Digging Deeper into …. Weight (all classes)

Category : Uncategorized

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 5og…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.

 

Further Reading & Viewing:


Maths by Month – March

Category : Uncategorized

In this March overview for Operation Maths users, there are links to topic-specific posts and articles, as well as a whole host of extra suggestions, links etc. To ensure you don’t miss out on any future Maths by Month blog-posts, please subscribe to the blog via email, on the top right hand of this page.

Pssst! The Edco Primary Publications launches for 2018 will be taking place in March and April in Kilkenny, Dublin North and South, Limerick, Cork, Galway, Athlone and Wexford. As well as launching Number Facts 1-4 and Bua na Cainte 3, they will also be showcasing Operation Maths and Exploring Spelling. Click on the link above for more info and to register.

 

Operation Maths Jr Infs to 2nd classes:

  • Junior Infants will be exploring representing and interpreting Data (week 1), exploring aspects of Shape and Space via 2D shapes, 3D shapes and spatial awareness (week 2), and developing an understanding of the concept of Time via sequencing of familiar daily events and the days of the week (week 3). As usual, a whole suite of supporting activities are detailed in the TRB as well as ideas for Aistear play.
  • Senior Infants will be exploring Length (week 1), as well as being formally being introduced to the number 10 (week 2) and then exploring 10 further by investigating the different combinations that make up the number 10, using a variety of models including the Operation Maths ten frames (week 3). They will also further consolidate their understanding of patterns (Algebra) and graphs (Data) via linked activities. As mentioned previously, the  Number Talks resources  for the number 10 available at the link are very applicable as well as the practical activities and stations using concrete materials, that are suggested in the TRB.
  • First Classes will be looking at Length, Operations and Fractions.

In Operations, the children will be exploring addition and subtraction without renaming. On many of the pages in the At School books the calculations are presented horizontally. This is deliberately done to encourage the children to complete the calculations using concrete materials, pictorial representations and/or mental strategies, as opposed to always using the vertical column method. While the development of traditional written procedures (eg the column method) is still important, these written methods are not more important than the development of mental computation skills and the ability to visualise and manipulate numbers mentally.

Some of the mental strategies for addition and subtraction, used in Number Talks, would also be every useful here e.g. partitioning, friendly numbers, making tens (compensation) and removal (deduction). Suitable number strings can be accessed at the link above and of course teachers can make their own number strings based on the horizontal calculations in their At School book, for example, from page 93 (four calculations in each number string works well) : 39 – 7, 39 – 0, 39 – 1, 39 – 9

  • Second Classes will be spend most of this month (week 1 & 2) working on operations; addition, without and with renaming,  and subtraction without and with renaming, the latter of which is new content while the other material will be revision of first class work. Concurrently, they will will also be revising place value to 99 as the basis for the work on the operations.On many of the pages in the At School books the calculations are presented horizontally. This is deliberately done to encourage the children to complete the calculations using concrete materials, pictorial representations and/or mental strategies, as opposed to always using the vertical column method. While the development of traditional written procedures (eg the column method) is still important, these written methods are not more important than the development of mental computation skills and the ability to visualise and manipulate numbers mentally.

    Some of the mental strategies for addition and subtraction, used in Number Talks, would also be every useful here e.g. partitioning, friendly numbers, making tens (compensation) and removal (deduction). Suitable number strings can be accessed at the link above and of course teachers can make their own number strings based on the horizontal calculations in their At School book, for example, from page 81, four new calculations could be made from one (four calculations in each number string works well) : 56 +16, 56 +36, 56 +19, 56 +27; 94 – 5, 94 – 15, 94 – 26,  94 – 38.

For second class, week three will be concerned with Length and the formal introduction of the centimeter as a standard unit of measure. 

(click on any of the links above for more information)

  • Operation Maths users can also access a class specific, month-by-month list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 
  1. Log into your edcolearning account
  2. Click on the At School Book for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select Weblinks from list of categories and then click to download the document.
  • Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above.

 Operation Maths 3rd to 6th classes:

The topics for this month are:

Click on each link above to access more in-depth information and links on each of the topics for this month.

  • Operation Maths users can also access a class specific, topic-by-topic list of relevant links and online resources via the Weblinks document, accessible on www.edcolearning.ie. 
  1. Log into your edcolearning account
  2. Click on the Pupil’s Book for your class level.
  3. Click on the Edco Resources icon (on book cover image on left-hand side)
  4. Select Weblinks from list of categories and then click to download the document.
  • Also accessible on  www.edcolearning.ie.  are the custom-made digital resources to support these topics. These will all be viewable when you click on the Edco Resources icon as directed above. 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.

Digging Deeper into … Symmetry (2nd to 4th)

Category : Uncategorized

Symmetry is officially a strand unit for second to fourth classes, although it also features as a content objective in 2-D shapes for fifth and sixth class where the children “classify 2-D shapes according to their lines of symmetry”.

While there are different types of symmetry, the curriculum specifies line symmetry, also known as mirror symmetry, reflective or reflection symmetry.

In Operation Maths, this chapter is placed after 2-D shapes, so that the children can identify symmetry in the shapes that they have previously encountered, and, in third  and fourth class, it is placed after Lines and Angles so that they can use their knowledge of different line types when describing the lines of symmetry.

Concrete exploration

To complete or create symmetrical patterns, requires the children being able to visualise the mirror image of the given arrangement/image. But children cannot visualise what they have not experienced. Thus to experience symmetry the children must:

  • be made aware of examples of symmetry all around them, and locate examples themselves e.g. flowers, leaves, objects at home and at school, numbers and letters of the alphabet.
  • be afforded ample opportunities to use real mirrors to explore symmetry. The type of child-safe mirrors that are often used in science investigations (eg in the strand unit of light) are ideal for this purpose.

Using mirrors allows the children the opportunity to observe symmetry and to check the accuracy of their completed patterns.  When using mirrors:

  • Try to have enough mirrors for one between two (the child-safe mirrors can often be cut into smaller sizes, 10cm x 7cm approx is big enough), or if supply is limited the mirror exploration could be incorporated as a station in a station/team teaching maths lesson.
  • Initially, allow the children free exploration and then, when suitable, guide it towards a purpose using questioning:
    • What letters or numbers look the same in the mirror? What shapes or images in the environment look the same in the mirror?
    • Can you put the mirror along the middle of any shapes and numbers so that they look complete? Does this work with any shapes or images from the environment? Don’t specify “middle” as being horizontal or vertical, and then see if the children realise that, on some figures, there is more than one than one way that the mirror can be placed.
  • At this point you could use this as the introduction to a separate and distinct What do you notice? What do you wonder? activity, and use the children’s wonder questions to guide the course of the rest of the lesson.
  • Explain that, on the symmetrical figures, the position of the mirror, is referred to as the line of symmetry. Then ask the children to use the mirrors to identify/draw the line of symmetry on the figures or mark the line of symmetry first (more challenging) and then check using the mirror.
  • Using the mirrors the children can create and check symmetrical patterns using cubes, counters, objects etc. One child can create a pattern that their partner has to complete symmetrically. Since children often incorrectly replicate the pattern (eg as done in the first image below) rather than reverse it, the mirror can show them their error (as used in second image below). Encourage the children to realise that whatever is closest to the mirror/line of symmetry on one side will also be closest to the mirror on the other side.

  • The children could then progress to creating symmetrical arrangements of more than one row. The Operation Maths twenty frames (free with Operation Maths 1 and 2) can be very useful for this (see below). Again the children should be encouraged to recognise that the colour and type of object/figure that is closest to the line of symmetry on one side should also be closest to the line of symmetry on the other side.

When the children have had sufficient experience with actual mirrors they should progress to completing activities without them, although they could always be returned to again if needs arose.

Other Resources

 


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.

 

Further Reading & Viewing:

 


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