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6 Subtraction Strategies to Help Students Get Comfortable With Mental Math

Written by Jeanne Sager

Ah, subtraction strategies. Whether you’re new to teaching subtraction or looking for some new methods to add to your teacher toolbox, teaching young mathematicians different strategies to help them tackle a problem will not only boost their confidence in their mathematic skills now but also serve them as they grow and move on to more complicated math.

The math teachers on the teacher team at Teach Starter have taught a variety of subtraction strategies in our years in elementary classrooms, and we’ve put together this guide to help whether you’re brand-new to this part of the curriculum or just haven’t taught subtraction in a while. Feel free to explore and choose the strategies that work best for you and your learners.

Why Do Kids Need Subtraction Strategies?

But wait a second, do students really need a bunch of strategies to subtract numbers? Is that really the best way to make sure kids meet the core standards? We assure you, there is a method to this madness!

Subtraction can be particularly challenging for young students, and that can be frustrating for teachers too. But this is not a knock on your abilities as a teacher. Research has long shown that young kids find subtraction harder than addition and some experts have called for less emphasis on subtraction fluency in the primary grades in favor of making sure students have a solid understanding of addition first.

Need subtraction worksheets? Get started with teacher-created options here ! 

Researchers have also proven that students have substantially more success with subtraction when they can use their addition skills. Likewise, learners who understand the inverse relationship between addition and subtraction demonstrate better conceptual knowledge and — you guessed it — fare better with subtraction.

That’s where subtraction strategies play an important role in teaching this particular arithmetic operation. Rather than having to remember all their subtraction facts, students employ reasoning and their understanding of number relationships to come up with answers to the problems before them.

We know everyone learns differently, and students need to learn a variety of strategies as they develop their math fluency, so we’ve pulled together a host of options to help your students be more successful at subtraction.

Subtraction Strategies

Remember what we said about kids who understand the inverse relationship between addition and subtraction? The Make Ten subtraction strategy makes use of that relationship by reminding students that any subtraction problem can be reformed as a missing addend problem.

Many teachers teach students to break apart the larger number in a subtraction problem to “make 10.”

For example, say a student needs to solve 14-9. You might ask the student, “How can you change 14 into a ten? Take away 4.  This becomes the first number in your number bond for the number 9.  How many more do we need to make 9? (4 + ___ = 9) Five. Five is your answer.”

Tip: Using ten frames and counters will help students to understand this strategy at first before they can move on to breaking down the math mentally. Download a fun ten frames match-up game !

If the numbers are near one another, students can simply “count up” from the number being subtracted (subtrahend) to the whole (minuend). This works best with numbers that are within 10 digits such as 456 and 459 or 21 and 27. Using word problems can help cement this particular subtraction skill.

Use Doubles Facts

Another subtraction strategy that makes use of addition, using doubles plays off student memorization of doubles facts. For example, when a student remembers that 8 + 8 = 16, this can be used in reverse to deduce that 16 – 8 = 8.

Use a Hundreds Chart

Teaching students to subtract two-digit numbers? Keep hundreds charts handy in your classroom for this subtraction strategy!

Let’s say you need to subtract 57-29. Find 53 on the chart. You know that 29 has 2 tens and 9 ones.

Start at 57 on the chart, then go back 2 tens, moving you to 37. Then count back 9 ones, moving you to 28.

Snag an animated PowerPoint to use with students who are learning how to use the subtraction algorithm and borrowing.

Split Strategy

Another popular mental math strategy for solving subtraction problems is known as the split strategy because — you guessed it — it involved splitting one of the numbers in the problem! In this case, students will split the second, smaller number (the subtrahend) into expanded form. This strategy will work with various subtraction problems — even those that need regrouping (more on that later).

So how does it work? Let’s say you have the subtraction problem 164-48.

The student will break the 48 down into its place value values or expanded form — 40 and 8.

Next, they’ll subtract 40 from 164, which can be done by counting back in 10s from 164 to 124. Finally, they subtract 8 to find the answer, 116.

This subtraction strategy is also often called the “break apart” strategy as you’re breaking apart your subtrahend into the tens and ones.

Using a Number Line

Some students are visual learners, and using a number line is a subtraction strategy that plays to that strength! Sometimes called the “jump strategy,” this method allows students to subtract smaller numbers very easily. Students find the minuend on their number line and move toward the left according to the number given as the subtrahend, counting as they “jump” the space between them.

As students develop in their skills, they can progress from counting the difference out by 1s to counting by 5s or 10s.

Tip: Provide each student with a number line that they can keep close by, along with a dry-erase sleeve to make the number line easily re-usable. Students can use a whiteboard marker to “jump” along the line to solve a variety of subtraction problems.

Get more subtraction strategy tools to teach this building block of math in your classroom , curated by the Teach Starter team of expert teachers.

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Browse by subject and age group, subtraction grids.

Can you meet the challenge to see how many subtraction calculations you can do in two minutes? There are different levels and you can choose either one or two missing numbers to make your number sentence correct.

Subtraction to 10

This game provides a simple introduction to the concept of subtraction. Each calculation features a subtraction within 10, along with a pictorial representation of the start number.

Mental Maths Train

Mental Maths Train is a maths game which focuses on the essential vocabulary of addition, subtraction, multiplication and division. A multiple choice game which can give children confidence in the four arithmetic operations.

Coins game is a money game which introduces children to coinage in British, Australian, American and Euro currencies. There are three modes: Sorting, Ordering and Counting money.

Elmo's World Games

Have fun exploring Elmo's World where there are lots of activities for you to try. Some games are useful for learning early shapes.

Subtraction using concrete and pictorial strategies

A video lesson which shows how to solve subtraction problems using counters and pictures. It demonstrates how to set up and solve the subtraction questions and gives ideas for a subtraction game at home.

Fruit Splat Subtraction

Splat the fruit with the correct answer to the subtraction questions. There are 7 levels of difficulty including subtraction of two-digit numbers with regrouping.

Toy Shop Money Game

Money games where children work out which coins will buy toy shop items and where they can calculate change. Tablet-friendly and with different levels of difficulty. GDP, USD, EURO and AUD options are available.

Catch the Stars - Subtraction

Catch the stars in your bucket by working out the subtraction sums. Either subtract by mixed numbers or a single number up to 10. Be careful not to miss the stars.

Robot More or Less

Robot More or Less is a flexible addition or subtraction matching game. It is useful for developing mental maths skills.

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Free EYFS Practical Subtraction IWB Activity

This EYFS Practical Subtraction IWB activity includes questions that enable children to develop their understanding of subtraction. Children need to knock the bottles off the wall to help solve the subtraction calculation. Children will then discuss a true or false statement about subtraction.

More resources for EYFS . Development Matters Statement Links 3 and 4 year olds (M-4) Know that the last number reached when counting a small set of objects tells you how many there are in total (‘cardinal principle’). (M-6) Link numerals and amounts: for example, showing the right number of objects to match the numeral, up to 5.

Children in Reception (M-21) Count objects, actions and sounds. (M-23) Link the number symbol (numeral) with its cardinal number value. (M-27) Explore the composition of numbers to 10.

Early Learning Goal - Have a deep understanding of number to 10, including the composition of each number. - Subitise (recognise quantities without counting) up to 5. - Automatically recall (without reference to rhymes, counting or other aids) number bonds up to 5 (including subtraction facts) and some number bonds to 10, including double facts.

This resource is available to play with a Taster subscription.

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EYFS Maths Summer Week 3: Addition and Subtraction

Eyfs maths mastery scheme of work.

Boost learning and help children on their journey to mastery with our EYFS maths scheme of work.

In this lesson – EYFS Maths Summer Week 3: Addition and Subtraction – you can find everything you need to teach maths with ease in the summer term of Reception. This pack contains:

– detailed lesson plan including small steps, vocab, key questions, whole-class teaching activity, adult-led activity, problem solving activities and continuous provision enhancements

– EYFS maths lesson PowerPoint, linked to the planning

– differentiated worksheets, linked to the planning

Explore our EYFS maths mastery scheme of work .

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Mathematics in the EYFS: addition and subtraction - Plus and minus

One way that children learn about addition is through practical experience. In everyday life, we add when we combine two sets of objects, such as opening a packet of six cakes and a packet of four cakes and putting them on a plate. Sometimes we then count to find the total; in general, we find the total mentally. Experience shows children that six cakes and four cakes always make ten cakes.

In subtraction too, children learn by practical experience. They learn that removing a particular number of things from a group always leaves the same number of things. And again, this helps them learn about numbers in the abstract. A set of six cars with two removed always leaves four so, eventually, they learn that 6 minus 2 is 4.

Addition and subtraction

How children learn

Children learn about addition and subtraction through meeting simple number problems as part of their daily life and by being encouraged to talk through how to find the solution.

Children's experiences of playing games that involve winning or losing a number of objects as they progress around the board or move along a track give them opportunities to be involved in doing lots of adding and taking away in a fairly short period of time. Playing in a group or family situation lets them see how adults or older children do addition and subtraction.

Many children begin to understand a notion of addition as they learn clapping songs, and finger rhymes that they have to act out first with the right hand and then the left and then both hands together.

Helping children learn

- To help children learn about addition, provide plenty of practical experience - games as well as real addition situations - involving objects, people, pictures and sounds. Introduce all the relevant language to help children talk about what is happening.

- To help children learn about subtraction, provide practical experience involving sets of objects, people and pictures where some objects are hidden or removed. And, again, model the relevant language for children to acquire for themselves.

- How many do you think there will be altogether?

- I wonder how many there are left.

- Have you got fewer pencils than I have?

- There were three dinosaurs in the box and two have escaped. How many are still in the box?

- I think five fingers and five fingers together makes ...

- If there were eight frogs sitting on a log and one jumped off, how could you work out how many were left sitting on the log?

- Have you got more or fewer pennies than you started with?

- Ask children to roll two different coloured dice together and to collect that many counters. Then discuss which colour counters there are more of.

- Ask children to dig up buried treasure from the sand tray with a friend and work out how many things there are altogether.

- Encourage children to make a ten-bead necklace and decide how many of each bead they need.

Assessing progress in addition and subtraction

- Joins in number rhymes and songs about increasing and decreasing numbers of fish, birds, sausages

- Responds to the instruction to take more counters, bricks, or buttons, or to remove some

- Uses counting to find the new number of items in a small group after some have been added or removed

- Uses counting to find the total number of items in two separate groups

- Uses counting to find the number left when some have been removed from a larger group

- Predicts results before they know the actual answer to an addition or subtraction situation: 'I think it will be seven'

- Can answer questions such as: 'There are five bugs in the box and two have escaped; how many are left?'

- Can mentally add 1 or 2 or subtract 1 or 2, to or from a number up to 10 or 20

- Knows all the number bonds up to 5 or more: 3 and 2 makes 5

- Knows that when a group of three or four beans is split up, the total is still the same.

Child-initiated play

Yours and mine Provide a selection of small, interesting baskets, bags or trays and interesting collections of items to fit inside them. Invite the children to choose a container and some items to put in it. Encourage them to talk about and compare their container with that of a friend. Develop the activity by playing 'Give again' with two or three children. The children take turns to give one or two things from their collection to the person sitting next to them. They count how many they have left or have gained each time.

Trams Use two shoeboxes decorated to resemble a two-carriage tram and make a queue of small-world characters waiting for the tram. The children decide how many should go in each carriage. Develop the play by adding more characters and another two-carriage tram.

Take some away Provide a set of small, lidded containers and fill half of them with different objects. Leave the other half empty so that the children can remove some objects from one container and put them in an empty container.

Adult-led activities

Chips for everyone Give everyone two paper plates. Take it in turns to take one scoop of chips (counting sticks) from the pan and put some chips on each plate. Help children decide if the two plates have the same number of chips on them or not: 'I think there are more chips on this plate. What do you think?'

Magic purses game Each child has a purse and there is a bank of pennies on the table. Everyone puts four pennies in their purse. A magician puppet says how many pennies to take out of the purses and then guesses how many are left in each purse. The children empty their purses and check. 'How many do you think are left? How many did we take away?'

Music area Provide instruments that make distinct sounds (drums, tambourines, chimes), chalks and boards. Together use two different instruments to make up simple five-beat tunes, such as two beats on the drum and three beats on the tambourine.

Find a way to record the music and display it for others to try. 'Can you remember how that music went? Can we work out how to play this on the board?'

Outdoor area Provide a collection of cardboard boxes, a soft ball tied with string attached to a washing line, easel and paper for scoring. Build a wall with cardboard boxes and swing the ball at the wall twice. Each time, count how many boxes were knocked down.

Record the total on a score board. 'How did you find out how many boxes there were to start with? Do you know a way to work out that total?'

IMPORTANT WORDS AND PHRASES

add, take away, leave, make, sum, total, altogether, enough, not enough, double, more, less, count on, How many more to make ...?, How many are left?

ASSESSING CHILDREN'S DEVELOPMENT

- comments that they and a friend 'both have the same' when each child has two pieces of apple

then they may be on this step

- compare two small groups of objects, saying when they have the same number

- show curiosity

- responds to the instruction to take one or two more bricks, or to give one or two coins to a friend

- counts up to three or four objects, even when these are not all together, such as two cherries on a plate and another cherry on the table

- can help you count four bears, and say there are still four bears, even when you move them around

- show an interest in number problems

- separate a group of three or four objects in different ways, beginning to recognise that the total is still the same

- adapt their behaviour to different events

- counts the paint brushes in the jar, in order to find the total number when some more have been added

- counts the paint pots on the table in order to find the number left when some have been removed

- find the total number of items in two groups by counting all of them

- use own methods to solve a problem

- show confidence

- With five counters, can tell you how many they would have if you gave them three more

- With six counters, can tell you how many they would have left if you took two of them away

- Can find out how many bears there are in two houses by counting both sets

- relate addition to combining two groups

- relate subtraction to taking away

- use developing mathematical ideas and methods to solve practical problems

- maintain attention and concentrate

- Finds out how many there are altogether by combining two lots of objects and counting them

- Takes away by removing six objects from a set and counting what remains

- Solves imaginary problems set in a familiar context

- Seems to just know some number bonds

- use a range of strategies for addition and subtraction, including some mental recall of number bonds

- sustain involvement and persevere, particularly when trying to solve a problem or reach a satisfactory conclusion

ABOUT THIS SERIES

This series aims to:

- build on practitioners' knowledge of how children acquire numeracy skills

- offer ideas on how to help children develop these skills.

Each part will focus on an aspect of numeracy included within the 'problem-solving, reasoning and numeracy' area of learning in the EYFS.

Authors Sheila Ebbutt and Carole Skinner are managing director and product development manager respectively of BEAM, which is dedicated to promoting excellence in mathematics education.

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Mathematics in the EYFS: Calculating - More or less

Mathematics in the eyfs: counting - in total, mathematics in the eyfs: problem-solving - within reason.

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Eyfs Subtraction worksheets

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25 Addition and Subtraction Word ProblemsYear 2 to Year 6 With Tips On Supporting Pupils’ Progress

Steven Eastes

Addition and subtraction word problems are synonymous with primary maths. Children encounter both operations very early on in their primary school journey. It is the understanding of both addition and subtraction facts that opens up a world within maths for children to tackle different mathematical concepts. For this reason, along with place value , understanding and applying these operations really does form the foundations of maths.

What are addition and subtraction word problems

Addition and subtraction word problems in the national curriculum.

• Why are word problems important for childrens’ understanding of addition and subtraction

How to teach addition and subtraction word problem solving in primary school

Addition and subtraction word problems for year 2, addition and subtraction word problems for year 3, addition and subtraction word problems for year 4, addition and subtraction word problems for year 5, addition and subtraction word problems for year 6, looking for more resources.

In Early Years (EYFS) children will be introduced to simple addition and subtraction word problems and they will visually explore how the total gets bigger when adding and how the total gets smaller when taking away. These skills, whilst basic, are built upon year on year up to Year 6 whereby children will encounter mixed word problems and complex word problems that involve two-steps in order to solve them.

All Kinds of Word Problems Addition and Subtraction

Test your students' problem solving skills with this pack of addition and subtraction word problems worksheets.

But without forming this solid basis in EYFS and Key Stage 1 where children explore through the use of visuals, manipulatives and concrete materials, children will not be as confident with the written formal methods and solving complex multi-step word problems involving large numbers and decimals.

Addition word problems and subtraction word problems are questions involving both operations, placed into context and given a real-life scenario. Rather than practising fluency through arithmetic, children are given math word problems to challenge them to interpret what a question is asking and what operation or multi-steps they need to do in order to find the solution.

In this blog we have put together a range of questions from Year 2 up to Year 6 to show the journey from one step word-problems, on to two-step problems and ending with more complex word problems that mimic those seen in the KS2 SATs. 

Addition and subtraction word problems in Year 2

Having already been introduced to simple addition and subtraction word problems worksheets in Year 1, where the focus is on representing simple number sentences visually, in Year 2 children continue to use pictorial representations and concrete materials to solve addition and subtraction problems.

Third Space Learning’s online one-to-one tutoring programme works to ensure children understand mathematical concepts by for example, using pictorial representations.

The word problems will typically focus around number, measures and simple money word problems e.g. adding coins together. Children will also be building their mental methods in calculating both addition and subtraction and be introduced to the different vocabulary both operations through quality first language and flashcards in lessons. By the end of Year 2, children will be expected to solve word problems using 1 digit and 2 digit numbers and will be introduced to the concept of regrouping and using a number line to solve the problems.

Addition and subtraction word problems in Year 3

Word problems for Year 3 signal the beginning of solving more complex word problems. Children begin to use more formal written methods and go beyond using a number line, although this may be applicable to some questions.

In Year 3 children begin to use column addition or subtraction to solve problems involving 3-digit numbers and their knowledge extends to adding and subtracting word problems involving topics such as fractions. They also consolidate their place value knowledge by being able to mentally add 3-digit numbers with ones, tens and hundreds. The concept of inverse is also important here to see the connection between addition and subtraction and how we can use the inverse to solve a problem within a maths activity.

Addition and subtraction word problems in Year 4

With word problems for Year 4 , children make the next progression to solving two step word problems and moving from 3-digit numbers to 4-digit numbers. When seeing word problems in context children will have to decide what operation to use and what method is most efficient and accurate for the problem. They also will need to and understand why. 

Whilst working with larger numbers pupils will also need to think about the processes and strategies behind multi digit addition and how to use subtraction facts to assist in solving mixed word problems.

Addition and subtraction word problems in Year 5

As children progress to Upper Key Stage 2, they will encounter two-step addition and subtraction word problems for Year 5 , they will be expected to solve contextual word problems by adding or subtracting both whole numbers up to 5 or 6 digit numbers, as well as decimal numbers.

 Whilst children will be increasing their ability to add larger numbers mentally, they will be tightening their grasp of adding and subtracting large numbers using the formal written methods. Children should also be encouraged to use the inverse, for example in subtraction word problems, addition can be used to check the accuracy of their answer and to consider how to effectively estimate the answer before proceeding with the formal written approach.

Addition and subtraction word problems in Year 6

In preparation for the SATs tests towards the end of Year 6, children will be solving increasingly more complex word problems in both addition and subtraction. Word problems will be given a real-life context and will involve large numbers in the millions, as well as decimal numbers. 

The multi-step word problems for Year 6 will also not be limited to purely addition and subtraction word problems but extend to using all four operations: addition, subtraction, multiplication word problems and division word problems and children will need to be able to interpret them successfully and apply their understanding in the correct manner.

Why are word problems important for childrens’ understanding of addition and subtraction

Word problems for addition and subtraction do not only bring the maths to life, they enhance understanding of both operations. From simple story problems where children use visuals and concrete materials to explain what is happening, to being able to solve complex multi-step problems, word problems go beyond a maths worksheet aimed purely at fluency in addition and subtraction.

Word problems from EYFS also help engage children in applying their phonics skills, build up their mathematical vocabulary, build up their problem solving skills and deepen their understanding of the inverse between addition and subtraction. 

Finally, the use of word problems also enables children to build up their strategies to solve such a problem: pictorial, mental methods or formal written methods. The key is to engage via the language arts and provide a context where children can decipher what to do, how to do it and apply their knowledge to real-life situations that make sense to them.

Building on the arithmetic and fluency of adding and subtracting, the next step is to use concrete materials and pictorial representations to show what maths is happening. 

Children need to be able to read a word problem and successfully interpret what is happening by reading the question carefully and knowing what is asked. From here, children need to work out what operation is needed to solve the problem and solve it using one of the strategies in their armoury – hopefully the most efficient one for that problem!

Below is an example of an addition word problem that also includes subtraction:

In a countryside farm there are 4672 birds nesting in the trees.

A further 304 arrive to nest too but 561 depart for a better habitat.

How many birds are there in the countryside farm?

How do we solve this?

Firstly, we interpret the question to work out what we know:

• We know that at the beginning there were 4,672 birds.
• 304 more birds arrive….so we add this to the total above. We then get a new total of 4,976.
• 561 leave…so we take this away from the new total of 4,976 and we get the final total of 4,415.

How could this look pictorially?

Using the bar model method (suitable for Years 3 and 4) we could firstly do a bar model to add the 304 birds to the initial amount of 4,672.

Then we can take that total to show how many birds were left after 561 departed.

Alternatively, we could use the column method to represent this word problem too. It would look like this:

In Year 2 pupils are required to work with both 1-digit and 2-digit numbers. This is a vital stage for children to visually understand the processes of both addition and subtraction and so using a template to represent the word problems pictorially is a great way of allowing children to show what they interpret the question to be asking, and how to resolve it.

Mrs. Molloy had 18 glue sticks in her class but 6 had to be chucked away as the glue lids had been left off. How many does she have left now?

Answer: 12 glue sticks

This could be solved in several ways. We could represent this pictorially with 18 glue sticks and cross out 6, we could show this on a number line and count back 6, or we could show it in the bar model method. Here are images of how this could look:

Pictorially:

Number line:

The school bus arrives at a bus stop and 5 children get on the bus. At the next stop, 7 more children get on and at the final bus stop, 4 more children get on the bus.

How many children are now on the school bus altogether?

Answer: 16 children

5 + 7 + 4 = 16

a) Sahana was collecting football stickers and counted 32 in total. Hannah had 25 football stickers too. How many do they have altogether?

b) If Sahana then gave 11 of her stickers to her friend Kevin, how many stickers does Sahana have left now?

Answers: a) 57 b) 21

In part one 32 + 25 = 57.

In part 2, if Sahana had 32 to start with and gives away 11 so this would be, 32 – 11 = 21.

Flowers come in bunches of 10. Miss. Spalding buys 3 bunches for her house. How many flowers does she have in total?

10 + 10 + 10 = 30 or a multiplication fact that 10 x 3 = 30

Who has the most books – class 1 or class 2?

Class 1 had 37 books and were then given 15.

Class 2 had 60 books but lost lost 6.

Answer: Class 2 had the most.

Class 1 is 37 + 15 = 52 books

Class 2 is 60 – 6 = 54 books

In Year 3, children are required to be confident at adding and subtracting 3-digit numbers and they will begin to use the formal method of column subtraction or addition. Children will also notice that the type of word problems are a little more complex in using measures, time, money as ways of incorporating an addition or subtraction word problem.

176 poppies were growing in a field.

Over the spring months another 210 poppies grew in the field.

How many were there at the end of spring?

Answer: 386 poppies

176 + 210 = 386

It may be worth informing pupils that unlike subtraction, it does not matter what way round we carry out the addition with two or more different numbers. For example, it doesn’t have to be the biggest number first, but it is essential we do this in subtraction.

We could represent this in simple column addition as such:

Alice has saved £120 from her birthday.

She spends £35 on a plushie.

How much does she have left?

Answer: £85

120 – 35 = 85. Here we do need to borrow from the tens column and give to the ones column in order for subtraction to be possible.

Ariana scored 156 Dojo points last term. This term she has scored 137 more. How many points has she scored altogether?

Katie has 50 points less than Ariana. How many did she have?

Answer: Ariana had 293 Dojo points altogether and Katie had 243 Dojo points.

First we have to add Ariana’s together: 156 + 137 = 293.

Secondly, we have to subtract 50 from Ariana’s total, so 293 – 50 = 243.

Year 3 were asked what their favourite fruit was between strawberries and bananas. If there there are 145 children in Year 3 and 72 like strawberries the most, what is the most popular fruit? Strawberries or bananas?

Answer: Bananas wins with 73 children preferring them to strawberries.

145 – 72 = 73.

If assembly starts at 9:00am and finishes at 9:15am, then maths is an hour and 10 minutes long before break time. How many minutes is it from assembly to break time?

Answer: 85 minutes. This could also be one hour and 25 minutes if asked to convert.

15 + 60 (one hour) + 10 = 85 minutes.

Children in Year 4 consolidate their knowledge of adding and subtracting with 3-digit numbers before advancing to 4-digit numbers. They will also begin solving more complex word problems and multi-step problems too. This is also the perfect opportunity to begin self-checking their answers by using the inverse as they will also encounter decimal numbers and fractions within word problems.

James and Bernard have 820 marbles between them.

However, James has 148 more than Bernard.

How many does Bernard have?

Answer: Bernard has 336 marbles.

Additional information: James has 484 marbles.

How to solve this:

As there are two people, James and Bernard, and between them they have 820 marbles, we halve that to work out how many they would have each if divided equally.

The answer would be 410 as 820 divided 2 is 410.

But James has 148 more, so we halve 148 and get 74. 

This is because we will then take 74 from Bernards 410 and they will go to James. 410 – 74 = 336.

So for Bernard we subtract 74 from the 410, and for James we add 74 to his 410. This gives a difference of 148.

I have £25.

I need to buy a birthday present for £6.50, lunch for £9.75 and a train ticket for £5.50.

Will I have enough?

Answer: Yes! Not only will I have enough, I will have £3.25 left.

25.00 – 6.50, 9.75 and 5.50 = £3.25

A private jet can hold fuel for 8,689 miles.

To travel around the UK, it will use 7,863 miles worth of fuel.

To get to France is another 710 miles. Will the private jet have enough fuel?

Answer: Yes because 8689 – 7,863 = 826

Therefore, we would still have enough fuel for 826 miles and France is under this at 710 miles.

A return ticket to Japan is £1550.

Jack has saved £865 and Olivia has saved £379. Have they saved enough?

If not, how much more do they need to save?

Answer: No they do not have enough. They need another £306.

865 + 379 = 1,244.

1,550 – 1,244 = 306

Team Allstars are training for the big race. They spend 1928 minutes training in February and then 5674 minutes training in March.

Their rivals, the Dream Team, spend 7642 minutes training in February and March combined. Who spent the most time training for the big race? What is the difference in training times?

Answer: Team Allstars spends 7,602 minutes training so the Dream Team spent longer training. The time difference was 40 minutes.

As children begin Upper Key Stage 2 they will become familiar working with larger numbers, 5 and 6 digit numbers, as well as decimals. Word problems are typically more complex and are mainly two-step problems together with the use of multiplication and division as well. They will again be encouraged to check the accuracy of their answers by using the inverse.

11,347 cupcakes were baked for an event. 8,692 were sold at £3 each.

How many cupcakes were left and how much money was made?

Answer: There were 2,655 left and £26,076 was made from selling the cupcakes.

11,347 – 8,692 = 2,655

8,692 x 3 = £26,076

A cruise ship had 12,469 passengers in April and 14,738 in May. How many more passengers were aboard in May?

Answer: 2,269 passengers

14,738 – 12,469 = 2,269

At the beginning of the school year, there are 11, 499 pencils. The school receives an extra 3,575 midway through the school year but they use 9,223 during the year.

Maria works out how many are left at the end of the school year by doing the following calculation:

9,223 + 3,575 = 12,798. Then 12,798 – 11,499 = 1,299 pencils left.

Explain the mistake Maria has made and work out how many are actually left at the end of the year.

Answer: Maria has completed this question incorrectly because she had to add the extra 3,575 to the initial 11,499. This would give her 15,074. She then had to subtract 9,223 from this to give the amount left at the end of the year which would be = 5,851

A car showroom sells 3 Mercedes sports cars for £72,731 each and 2 Porsche sports cars for £107,299 each. How much did they make in total? What is the difference between the total the Mercedes sports cars made compared to the total the Porsche sports cars made?

Answer: The Mercedes total cost was £218,193 and we can do this either by adding £72,731 three times, or multiplying it by 3.

The total cost for the 2 Porsche cars was £214,598. If we then subtract £214,598 from £218,193 we will get the difference which is £3,595.

Martin is trying to work out would be longer in length. Eight pieces of string that are 1.6m in length each or 12 pieces of string that are 0.8m in length. What is the total length of string if we added both of them together?

Answer: Eight lots of string at 1.6m each is 12.8m in total (either repeated addition or multiplying by 8) and 12 pieces of string at 0.8m are 9.6m in total. Altogether we add them up and get 22.4m.

In preparation for the SATs, children in Year 6 need to be able to add or subtract when using 6 or 7 digit numbers, as well as decimal numbers. Addition and subtraction word problems in the SATs may also include fraction word problems and they will be two-step or multi step word problems to interpret and solve.

A toy shop is holding a buy one get one half price offer on the new computer. The computer costs £290.50 for one. How much would 2 cost in the offer?

Answer: £435.75

To find the cost of the second one we divide £290.50 by 2 and get £145.25 and then add it to the cost of the original one.

£290.50 + £145.25 = £435.75

Sonic the Hedgehog 2 is playing at two different huge outdoor cinemas that both hold 250,000 people. One cinema has 218,745 people watching the film and the other has 187,681.

How many people are watching the film? How many more people could watch the film?

Answer: There are 406,426 people watching the film. 93,574 more people could watch the film.

This is: 218,745 + 187,681 = 406,426

93,574 more people could watch the film.

This is: 500,000 – 406,426 = 93,574

Jagat buys snacks for his friends and himself. When he weights the bag, it weighs 1.2kg but when he removes just the chocolates it weighs 990g.

When he removes just the crisps it weighs 830g.

How much more do the crisps weigh than the chocolates?

Answer: The chocolates are lighter as they weigh 210g. The crisps weigh 370g. So the crisps weigh 160g more than the chocolates.

Convert 1.2kg into grams = 1200.

Then 1200 – 990 = 210g for the chocolates

Then 1200 – 830 = 370g for the crisps

370 – 210 = 160g difference

Griffin is thinking of a number,

He adds 1,728.

He subtracts 352.

He divides it by 4.

His answer is 958.

What number did he start with?

Answer: 2,456

To do this we start with 958 and follow the rules backwards by doing the inverse.

958 x 4 =3,832

3,832 + 352 = 4,184

4,184 – 1,728 = 2,456

Ben bought 1.4 litres of orange juice for him and 15 friends to enjoy. How much juice would they all get?

If all 16 children wanted 100ml of juice each how much does each child need extra and how much juice would that be altogether in litres?

Answer: 87.5ml of juice each is the original serving.

They would need a further 12.5ml each to have 100ml in their cup.

12.5ml x 16 children would be an extra 200ml.

Add that to 1.4 litres it would equal 1.6 litres.

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Early Years Guide

Introduction.

The first few years of a child’s life are especially important for mathematics development . For many education experts, no other group represents a greater opportunity to improve mathematical standards than children in the early years.

The more grounded in mathematical concepts young children become, the better their later outcomes. Conversely, research shows that children who start behind in mathematics tend to stay behind throughout their educational journey.

On this page, we’ll examine:

• What do we mean by Early Years?
• What does learning look like in the Early Years?
• Why is Cognitive Load Theory so important?
• What mastery strategies are available for Early Years?

What do we mean when we talk about Early Years?

The UK government published the Statutory Framework for the early years foundation stage in March 2017. It sets standards for the learning, development and care of children from birth to five years old.

Areas of learning

The EYFS framework outlines seven areas of learning :

• Communication and language
• Physical development
• Personal, social and emotional development
• Mathematics
• Understanding the world
• Expressive art and design

Mathematics in EYFS

In the context of mathematics, the framework says children must be given opportunities to develop their skills in the following areas:

• Understanding and using numbers
• Calculating simple addition and subtraction problems
• Describing shapes, spaces, and measure

Revised guidance

The DfE published revised guidance in March 2021 to take effect in September 2021.

The mathematics component now incorporates many elements of the mastery approach.

Specifically, the revised framework says:

Children should be able to count confidently, develop a deep understanding of the numbers to 10, the relationships between them and the patterns within those numbers.

By providing frequent and varied opportunities to build and apply this understanding — such as using manipulatives, including small pebbles and tens frames for organising counting — children will develop a secure base of knowledge and vocabulary from which mastery of mathematics is built.

In addition, it is important that the curriculum includes rich opportunities for children to develop their spatial reasoning skills across all areas of mathematics including shape, space and measures.

It is important that children develop positive attitudes and interests in mathematics, look for patterns and relationships, spot connections, ‘have a go’, talk to adults and peers about what they notice and not be afraid to make mistakes.

Early Learning Goals

The latest framework has the following early learning goals for mathematics:

Children at the expected level of development will:

• Have a deep understanding of number to 10, including the composition of each number
• Subitise (recognise quantities without counting) up to five
• Automatically recall (without reference to rhymes, counting or other aids) number bonds up to five (including subtraction facts) and some number bonds to 10, including double facts

Numerical patterns

• Verbally count beyond 20, recognising the pattern of the counting system
• Compare quantities up to 10 in different contexts, recognising when one quantity is greater than, less than or the same as the other quantity
• Explore and represent patterns within numbers up to 10, including evens and odds, double facts and how quantities can be distributed equally

Reception class is the first year at primary school in England, generally for children ages four to five. Unlike every other school year, it’s not compulsory for children to attend Reception, though it’s a good way to introduce them to life at school.

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Learning in the early years

The first few years of a child’s life are especially important for mathematics development , says the National Center for Excellence in the Teaching of Mathematics.

Research shows that early mathematical knowledge predicts later reading ability and general education and social progress.

As young as eight months old, children are developing an awareness of number names , and include these in their speech, as soon as they begin to talk. As children listen to the talk around them, they are introduced to numbers through opportunities that occur in everyday life, and experience a variety of number rhymes. This supports their growing knowledge of number names.

According to the NCETM, there are:

Six key areas of mathematical learning

Cardinality and counting, composition.

• Shape and Space

Looking briefly at each in turn:

When children understand the cardinality of numbers , they know what the numbers mean in terms of knowing how many things they refer to.

Comparing numbers involves knowing which numbers are worth more or less than each other.

Learning to ‘see’ a whole number and its parts at the same time is a key development in children’s number understanding.

Developing an awareness of pattern helps young children to notice and understand mathematical relationships.

Shape and space

Mathematically, the areas of shape and space are about developing visualising skills and understanding relationships, such as the effects of movement and combining shapes

Measuring in mathematics is based on the idea of using numbers of units in order to compare attributes , such as length or capacity.

Learning to count in the early years is a fundamental skill and key to mastering mathematical concepts in the future, but there’s more to it than you might think, says Sabrina Pinnock, a primary school teacher in Yorkshire.

According to researchers Rochel Gelman and C.R. Gallistel, these are the steps needed to successfully count :

• The one-to-one principle: children must name each object they count and understand there are two groups: the one that has been counted and the one that hasn’t yet been counted
• The stable order principle: children must know how to count in the right order
• The cardinal principle: children need to understand the last number in the set is the total amount
• Counting anything: children need to realise that anything can be counted, not just objects that can be touched, but also things like claps and jumps
• Order of counting doesn’t matter: children need to understand that the order of counting in the set is irrelevant and will still lead to the same amount

Assessing children to find out which step they are struggling with is key to helping them overcome difficulties and become confident counters.

How do children develop counting skills?

Very young children start to count spontaneously and later begin to refine their skills by pointing their finger at the objects they are counting.

They will often try to get all the names of the numbers they know into their count as they pass their finger along the objects. They also reuse numbers. If they have not finished and they have used up all their known numbers, they will begin using the same numbers again. For example, a child might decide to count eight shells she collects at the beach. She might line them up carefully, tag numbers to them by pointing as she slides her finger along the shells, quickly counting out loud, “one, two, three, four, five, one, two, three, four, five, one, two, three.”

In their drive to make meaning, children are eager to experiment as they acquire new small bits of mathematical knowledge. It is extremely important to respect their developing understanding and not expect “perfect” counting sequences.

By valuing children’s partial understanding, children will develop enthusiasm for numbers and become confident mathematicians.

Activities to boost number sense in Reception Year

Children need lots of opportunities to develop number sense and deepen their conceptual understanding. Here are some simple activities to get your Reception Year learners counting:

Crowd control

Display the number of children allowed in each area using pictorial representations of cubes on a 10 frame. Once the children begin to realise how many are allowed in the area, they start to discuss the meaning of more and less. For example, “no more children are allowed in,” or “you can come in because one more than three is four.”

Encourage children to show numbers using their fingers above their head. “Bunny ears six” means they place their fingers above their head to show six. They may decide to use three fingers on each hand. As they become more confident, you could introduce swapping, where they show the same number but with a different configuration of fingers, in this case two and four, or five and one.

Grouping straws

Each morning, drop different amounts of art straws all over the carpet. Say something like, “oh no class, I can’t believe it. I’ve dropped all my straws again. They were all in 10s. Can you help me?” This activity helps children consolidate counting objects and gets them to think about stopping after they have made 10. Providing elastic bands helps them to keep track of their groups of 10.

Fastest 10 frames

This game can help distinguish between those who have developed a good understanding of number sense and those who need further support. Give each child their own frame and cubes. Tell them a number and observe how they place the cubes on the frame. If the children are working with the number eight, do they say each number name as they place the cube on the frame, or do they realise eight is two less than 10? If so, they should be able to place the cubes down faster than other children.

What do they do when you say the next number? For example, for the number five, do they automatically remove three cubes, or do they remove all of the cubes and start over counting from one to five?

Everyday questions to develop number sense

These questions for children aged five to six help develop their number sense and let them practice using mathematical terms.

When prepping lunch or a snack, count out the different types of food with your child, and as you lay the table, count out the different items. Ask your child questions like:

• How many grapes are there?
• How many tomatoes are there?
• How many plates are there?

Practice using the terms more than, fewer than and as many as by asking:

• Are there more grapes than tomatoes?
• Are there fewer tomatoes than grapes?
• Are there as many plates as people eating?

Remember to practice each sentence:

• There are more grapes than tomatoes
• There are fewer tomatoes than grapes
• There are as many plates as family members eating

When counting, make sure that you count one number for one item to strengthen your child’s sense of one-to-one correspondence.

Number Rhymes

Carefully select number rhymes to include those that children are familiar with from home. Make sure the rhymes include:

• Counting back and counting forward
• “No” or “none” (Five little ducks went swimming one day)
• Counting in pairs (two, four, six, eight, Mary at the cottage gate)
• Counting to five, 10 and beyond

Problem solving, reasoning and numeracy

The EYFS requires children to be supported in developing their understanding of problem solving, reasoning and numeracy in a broad range of contexts in which they can explore, enjoy, learn, practise and talk about their developing understanding. They must be provided with opportunities to practise these skills and gain confidence.

Young children learn best through play. For their learning to be effective, they need sensitive and informed support from adults.

All children can be successful with mathematics, provided they have opportunities to explore ideas in ways that make personal sense to them and opportunities to develop concepts and understanding. Children need to know that practitioners are interested in their thinking and respect their ideas.

Foundations

Maths — No Problem! Foundations is designed with all the theory and rigor that underpins a true mastery approach. It meets all the requirements of the national curriculum’s Early Years Foundation Stage. But Maths — No Problem! Foundations doesn’t shy away from embedding learning through play in Reception.

Genuine learning through play in the early years is something the team at Maths — No Problem! gets very excited about. What may appear to be simple games are actually carefully designed activities that have a deep maths mastery focus.

Maths — No Problem! Foundations is a complete Reception programme that includes Workbook Journals, Picture Books, and online Teacher Guides with printable resource sheets, all in one package.

The Maths — No Problem! suite of products — including textbooks, workbooks, a revolutionary online assessment tool, world-class teacher training, and much more — is based on the Singapore method, which combines 30 years of international research with painstaking craftsmanship and constant refinement.

Mark making

Research from Carruthers and Worthington into children’s mathematical graphics reveals young children use their own marks and representations to explore and communicate their mathematical thinking. These graphics include:

• Scribble-marks
• Tally-type marks
• Invented and standard symbols including numerals

Young children’s graphical exploration “builds on what they already know about marks and symbols and lays the foundations for understanding mathematical symbols and later use of standard forms of written mathematics,” the researchers said.

In a 2009 publication, the UK Department for Children, Schools and Families, says practitioners should: “Value children’s own graphic and practical explorations of problem solving” and observe “the context in which young children use their own graphics.”

Developing understanding with careful questioning

When children play and interact with other children, there are always opportunities for maths talk to help them develop a deep understanding, says Sabinra Pinnock.

For instance:

• I have made a pattern. What’s your pattern?
• How many blocks taller is my model compared to yours?
• How do we know this area is full?
• I have three cars, how many do you have?
• Do you have more?
• How do you know?

Give learners long enough to think about their answer and give their response, but not so long that it disrupts the flow of play.

Adding maths talk activities to your daily routine

Developing maths talk in your daily routine gives learners a chance to understand concepts while using real-life concepts. It also means that children can consolidate what they have learned.

The following activities can get you started:

How many children are at school?

Get your class to work out how many children are at school by placing a picture of themselves or a counter representation on large 10 frames. Ask them questions like:

• How do we know this 10 frame is full?
• How many children are absent?
• What can you tell me about number seven?

Sorting and grouping objects as a class

Sorting and grouping objects as a class helps children learn to reason and look for patterns. Give them a variety of buttons each day and ask open-ended questions like, “how can we sort the buttons?” They can use critical-thinking skills to come up with a range of ideas like sorting by size, colour, pattern, and shape.

Vote for a story

First, ask a child to pick two books. Everyone in the class gets to vote (using a piece of lego, for instance) on which of the books should be read. Tally the votes at the end of the day to determine the winner. This can lead to questions such as:

• How many more votes did one book have than the other?

The key to introducing mastery in the early years is to keep activities fun and part of your daily routine. The more learners explore maths through play, the more engaged they become.

Pattern Awareness

Dr. Sue Gifford, emeritus fellow at University of Roehampton, says recent research shows a child’s ability to spot mathematical patterns can predict later mathematical achievement, more so than other abilities such as counting. It also shows pattern awareness can vary a great deal between individuals.

Australian researchers, Papic, Mulligan and Mitchelmore have found pattern awareness can be taught effectively to preschoolers, with positive effects on their later number understanding.

Explicitly teaching pattern awareness links to encouraging “pattern sniffing” with older children in order to develop mathematical understanding and thinking.

What is mathematical pattern awareness?

Patterns are basically relationships with some kind of regularity between the elements. In the early years, Papic et al suggest there are three main kinds:

• Shapes with regular features, such as a square or triangles with equal sides and angles, and shapes made with some equally spaced dots
• A repeated sequence: the most common examples are AB sequences, like a red, blue, red blue pattern with cubes. More challenging are ABC or ABB patterns with repeating units like red, green, blue or red, blue, blue
• a growing pattern, such as a staircase with equal steps

Children who are highly pattern aware can spot this kind of regularity: they can reproduce patterns and predict how they will continue.

Why is pattern awareness important?

Spotting underlying patterns is important for identifying many different kinds of mathematical relationships. It underpins memorization of the counting sequence and understanding number operations, for instance recognizing that if you add numbers in a different order their total stays the same.

Pattern awareness has been described as early algebraic thinking, which involves:

• Noticing mathematical features
• Identifying the relationship between elements
• Observing regularities

The activity Pattern Making focuses on repeating patterns and suggests some engaging ways of developing pattern awareness, with prompts for considering children’s responses. Children can make trains with assorted toys, make patterns with twigs and leaves outside or create printing and sticking patterns in design activities.

Repeating Patterns

It is important to introduce children to a variety of repeating patterns, progressing from ABC and ABB to ABBC.

Focusing on alternating AB patterns can result in some young children thinking that ‘blue, red, red’ can’t make a pattern. They say things like, “That’s not a pattern, because you can’t have two of the same colour next to each other.”

Cognitive Load Theory

Cognitive Load Theory has gained a lot of traction in recent years as educators embrace evidence-based research to inform their evolving practice, says Ross Deans, a KS2 teacher and maths lead in Bournemouth, England.

What is Cognitive Load Theory and why is it important?

Why are new teachers so overwhelmed by tasks that more experienced teachers can juggle alongside multiple other responsibilities?

The answer is simple — new skills demand more attention.

This logic can be applied to any situation. When learning to drive, for example, you focus carefully on every small detail. That mental exertion can be very demanding. Compare that to the feeling of driving after you’ve been doing it for years; you may barely remember the drive, the process is so familiar.

Now put yourself in the shoes of your pupils. Each lesson provides fresh learning and new skills to master. Consider what happens inside your learners’ heads when they encounter new information, new skills and new vocabulary.

Working memory

Cognitive Load Theory , originated by John Sweller, acknowledges that working memory is very limited.

Working memory is the information we hold in our minds while we’re learning. The number of things that we can keep in working memory at one time is approximately four, plus or minus one, and perhaps even less for children.

It’s important to keep this in mind when planning and delivering lessons. If our learners cannot balance more than four things in their working memory, then we need to be very careful about the information we choose to present to them.

Intrinsic versus extraneous load

Intrinsic load includes anything that is necessary to learn a desired skill. In other words, the essential stuff.

Extraneous load is anything that will detract from desired learning. In other words, the stuff that should be reduced as much as possible.

It can be tempting while teaching to embellish lessons with child-friendly imagery and gimmicks. While It’s important to foster enjoyment, we should avoid distracting learners from the essential components of a lesson.

Supporting the transition to long-term memory

While acknowledging the impact of Cognitive Load Theory, we can consider the following to support our learners:

Focused learning objective

First and foremost, we must have a very clear idea of what we want our learners to achieve. Keep the limitations of the working memory in mind and let this guide the content you choose to include in a lesson.

Activate prior learning

At the start of the lesson, you may choose to design a task that encourages learners to retrieve essential skills. This means their working memory can hold on to new learning during the lesson.

Present information clearly

Take time when designing lessons to make sure information is presented clearly. Avoid unnecessary extras which may detract from the learning goal. Keep slides clean and similar in style.

Avoid cognitive overload

In maths, problems are often detailed and complex. Consider breaking questions up into chunks so that learners can digest each part separately. By taking away the final question, you can make a maths problem goal-free.

Maths mastery for Early Years

Given the importance of developing sound mathematical understanding in the early years, the maths mastery approach can be especially useful, considering its focus on problem solving and whole-class learning.

Early Years and CPA

If you’re teaching the Concrete, Pictorial, Abstract (CPA) approach in the early years, it’s best to focus on C and P. Here’s how to use concrete and pictorial representations effectively.

The CPA model works brilliantly in the primary years but for the youngest learners, moving onto abstract concepts too soon causes difficulties. Spending as much time as possible with concrete objects and pictorial representations helps children master number skills.

By the time they reach Key Stage 2, children need to develop their understanding of numbers by being able to visualise what the concrete looks like in their heads. Therefore, it’s positive that the revised EYFS framework focuses on numbers just to 10, from 20 previously.

If learners develop a deep understanding of numbers to 10, their chances of understanding larger numbers increases significantly.

C is for concrete

Concrete is the “doing” stage. During this stage, students use concrete objects to model problems. Unlike traditional maths teaching methods where teachers demonstrate how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical (concrete) objects.

Spending time with real-life objects

The theorist Jerome Bruner stresses the importance of children spending time learning maths through tangible items. Spending lots of time using real-life objects, solving real-life problems, and manipulating abstract concrete objects (when ready) such as cubes and counters is essential in the early years.

Ideas include counting out fruit for snack time, comparing, sorting and counting a range of different buttons, pasta, and even ‘magic beans’ linked to specific topics.

Early years and number bonds

By mastering number bonds early on, pupils build the foundations needed for subsequent learning and are better equipped to develop mental strategies and mathematical fluency. By building a strong number sense, pupils can decide what action to take when trying to solve problems in their head.

How to teach number bonds

Children are usually introduced to number bonds through the Concrete, Pictorial, Abstract approach . Here’s just one way to introduce and teach number bonds.

Concrete step

Children start out by counting familiar real-world objects that they can interact with. They then use counters to represent the real-world objects. From here, they progress to grouping counters into two groups.

By putting five counters into two groups, children learn the different ways that five can be made. For example, 3 and 2 as illustrated below. With further exploration, children work out other ways to break numbers into two groups.

Pictorial step

Now that they understand the concept with hands-on objects and experience, children progress to writing number bonds in workbooks or on whiteboards. Early number bond explorations might simply reflect the two groups of counters that they created during the concrete step, along with other combinations.

Abstract step

With the concrete and pictorial steps done and dusted, children progress to representing abstract problems using mathematical notation (for example, 3 + 2 = 5).

Early Years and place value

Number and place value are foundational concepts for all mathematics learning. This means we need to address how to teach place value as early as possible so that pupils can secure their knowledge of the concept.

How do you develop an early understanding of place value in the primary school classroom? Let’s start by defining place value. It is a system for writing numerals where the position of each digit determines its value. Each value is a multiple of a common base of 10 in our decimal system.

Here are some teaching strategies I’ve found useful when helping learners develop an early understanding of place value.

Progress through concepts systematically

Developing an understanding of place value requires systematic progression. Each new concept should build on previous learning experiences so that pupils can gain deeper, relational understanding as they go.

This approach ensures knowledge is developed, refined and applied correctly as numbers become meaningful tools for solving problems rather than just a series of symbols on a page. Most importantly, this starts our learners on the path to becoming confident problem solvers and pattern spotters.

Use the CPA approach to establish meaning

The CPA ( Concrete, Pictorial, Abstract ) approach helps pupils connect a physical representation of a number (concrete manipulatives) to that same quantity as shown in drawings or graphics (pictorial), and finally to the actual written name and symbol for that number (abstract).

Concrete resources are meaning makers. They add meaning to abstract representations of numbers so that when learners progress to the abstract phase, they know what those numbers stand for, what they mean, and how they relate to each other.

If a pupil can identify the meaning of each component in a problem, they are far more confident in how they work to solve it.

Teach the ‘10-ness of 10’

At an early level, spend as much time as possible studying the numbers from 0 to 10, as understanding the 10-ness of 10 is crucial for maths attainment, and it cannot be rushed.

Once this understanding is locked-in, follow this with an introduction to number bonds. Start with the additive relationships between numbers less than 10, then progress to adding and subtracting up to 10. This ensures that learners see 10 as an important ‘base’ number in all of their future maths applications.

Progress to 20, then to 40

I make sure to take my time teaching 10 and teen numbers so that a solid understanding of place value with numbers up to 20 is properly established.

I then extend the place value concept by working with numbers up to 40 — followed by addition and subtraction to 40.

Because pupils have learned to make 10 and use number bonds, they are ready to begin working with multi-digit numbers and regrouping. Focusing on numbers to 40 while developing the concept of place value also allows learners to associate numbers with easily-managed, physical quantities (meaning makers).

Use base 10 blocks for 100 and 1000

The work we’ve done building a gradual understanding of place value will have prepared pupils to progress to three-digit numbers. So we can now move on to studying up to 100.

We start here by developing an understanding of numbers in multiple place value representations. For example, one thousand five hundred is 15 hundreds or 150 tens.

Once they get the hang of that, learners then sharpen their counting, reading, and writing skills for numbers up to 1,000. Moving into addition and subtraction with numbers up to 1,000 — with and without regrouping — is the next step.

Here is where our work establishing an early understanding of place value is key, because pupils will intrinsically know why these algorithms work for three and four-digit numbers. Base 10 blocks are a great tool to help solidify those earlier place value ideas when working with numbers up to the thousands.

Approach larger numbers the same way

The CPA approach is once again our answer to learning place value in larger numbers. Apply those skills and always be on the lookout for chances to extend number and place value concepts.

For example, you can identify and complete number patterns or find missing digits on a number line.

From there you can explore strategies for mental mathematics as well as addition and subtraction for numbers up to 10,000. Take learners even deeper by having them explore place value with an emphasis on multiplication, division, and decimals.

Mastering maths concepts like place value in the early years is not just key to success in the classroom. It prepares learners for a lifetime of deep mathematical understanding by giving them invaluable real-world tools like resilience and problem-solving ability.

And a confident problem solver in maths is a confident problem solver in life.

Well done on making it to the end of our Ultimate Guide to Early Years.

We’ve looked at the definition of Early Years and what the government recommends in its revised guidance, and we’ve taken a deep dive into some of the most-effective strategies for teaching mathematics mastery in the Early Years.

We’ve also discussed Cognitive Load Theory and what it means for teachers in the Early Years classroom.

If you’d like to learn more about Early Years, we recommend checking out the following links:

• NCETM: How Early Years children develop mathematical thinking (Podcast)
• NRICH: Early Years Foundation Stage Homepage
• The School of School: Episode 17 Play and early years (Podcast)
• Maths — No Problem! CPA approach

Also, don’t miss our other Ultimate Guides:

• The Maths — No Problem! Ultimate Guide to Maths Mastery
• The Maths — No Problem! Ultimate Guide to Assessment

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Early years addition and subtraction resources

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Tes maths for early years resources is home to an unrivalled range of teaching tools for addition and subtraction, including: - Eyfs addition and subtraction games - Early years addition worksheets Plus much, much more. Whatever you need, you're sure to find it amongst thousands of free and premium resources.

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Kindergarten Math Worksheets Addition 150 Questions & Answer Key Provided Save Teachers Time HWork

Kindergarten Math

Squishy subtraction

Christmas Colour by Number Math Activities Printables Worksheets _ Holidays & Winter Season 15 pages

Addition and Subtraction with Eggs KS1

Chinese New Year Colour by number, crafts, colouring pages, Cootie catcher, Zodiac animal And Math Activities Printables Worksheets - Lunar New Year, Spring Season and Holidays

Number Sentences within 10 with Marbles

Christmas Name Craft and Activity for Preschoolers - Holidays activities printables Worksheets BUNDLE

Christmas Colour By Number & Addition and Subtraction Math Activities Worksheets - Holidays & Winter Season Pixels Art

Printable Christmas Colour By Number for Kids - Holidays Math Activities Worksheets colouring pages Bundle

Christmas Colour By Number Math Activities Printables Worksheets Pixels Art - Holidays and Winter

Subtraction from 5 10 12 & 20 - 4 fun Maths Activities for KS1

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• Maths: Open-ended Investigations
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Reception Maths: Open-ended Investigations Mathematical Problem-solving

Problem-solving tasks develop mathematical skills and problem-solving tactics. These open-ended investigations for Reception or Early Years settings are designed to take advantage of outdoor learning environments, but many of them can be adapted to run inside.

Session 1 Shape

Open-ended investigative tasks provide fun, stimulating contexts in which children can connect previous knowledge with new situations, develop mental flexibility, practise mathematical vocabulary and reason mathematically.

Print the sheets and stick them up in suitable play areas. They provide stimulating questions that will enable adults in your classroom to facilitate good mathematical language and learning. Each illustrated activity comes with a list of skills practised that you can use for assessment.

Open-Ended Task

Shape Hunt By looking for and finding shapes, children gain an awareness of similarities of shapes in the environment. They match shapes by recognising similarities and orientation, show curiosity and observation by talking about shapes and begin to use mathematical names for shapes.

More Shapes By looking for and finding shapes formed by windows, children gain an awareness of shapes, practise matching them, and begin to use mathematical names for them. Use language such as ‘square’ to describe the shape of solids and flat shapes.

• More Shapes

Sorting While playing with and arranging twigs, stones, leaves, etc., children can be encouraged to take an interest in shape and space. They can talk about similarity and difference, while sorting objects. Developing mathematical ideas and methods can be used to solve practical problems.

Session 2 Position and Direction

Trails Remember… just about anything you do indoors in maths can be done outside. Some children ‘come alive’ once out of the classroom and may just surprise you with the observations they make or the learning behaviours they show.

Scooters, Bikes, Trikes Riding a scooter, bike or trike prompts counting, consideration of same and different, and position and spatial properties.

• Scooters, Bikes, Trikes
• The Mathematical Journey

Obstacle course Children use everyday language to talk about position, distance and time when running, or walking, an obstacle course. They compare quantities and objects and solve problems.

• Obstacle Course

Milk the Maths: Wellies Encourage children to use everyday language to talk about position whatever they are doing! Putting wellies away is a colourful opportunity.

• Milk the Maths

Session 3 Number and Shape

Holes When digging holes children can use number names in order in familiar contexts. They can use everyday language to talk about size, capacity, position, distance and time. Holes offer fun opportunities to compare quantities and objects and to solve problems.

The Mud Kitchen Ask children questions about shape, space and measure while exploring mud. Consider similarities and differences.

• Mud Kitchen

Planting and Gardening While working in a school garden, children can practise using numbers to identify how many objects there are in a set. They say and use number names in order in familiar contexts, and count everyday objects.

• Planting and Gardening

Hoist Playing with a bucket on a hoist, children can use numbers to identify how many objects there are in a set. They can use everyday language to talk about size, capacity, position, distance and time and compare quantities and objects and to solve problems.

Session 4 Number and the Language of Addition/Subtraction

Leaves When playing with leaves, children have opportunities to see that numbers identify how many objects there are in a set and to say and use number names in order in familiar contexts. They can begin to use the vocabulary involved in adding and subtracting. They can relate addition to combining two and subtraction to ‘taking away’.

Tin Can Alley Play with cans to explore number names in familiar contexts and to.count up to 10 everyday objects. Children can begin to use the vocabulary involved in adding and subtracting and to relate addition to combining two groups of objects and subtraction to ‘taking away’.

• Tin Can Alley

Sand to Sandpit Children can fill a sandpit (or move sand from one place to another) and count up to 10 everyday objects and begin to use the vocabulary involved in adding and subtracting.

• Sand to Sandpit

Logs Put logs onto a trolley and say and use number names in order in familiar contexts. Count and use vocabulary involved in adding and subtracting. Show curiosity and observation by talking about shapes. Begin to use mathematical names for shapes.

More Logs Playing with logs offers countless opportunities to practise counting! Children can also begin to use the vocabulary involved in adding and subtracting and to relate addition to combining two groups of objects and subtraction to ‘taking away’.

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