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# Mental Math Strategies Every Child Should Know From 1st To 5th Grade

When we think of mental math strategies, we are essentially thinking about those math skills we can do in our heads, without using the formal written methods that we would use for longer questions and standard algorithm methods.

In this article we introduce you to some of the mental math strategies you can teach your students to help develop their mental math skills throughout elementary school. The Ultimate Guide to Problem Solving Techniques Download Free Now!

### What are mental math strategies?

Mental math strategies are accepted ways of working math out in your head, that help us take shortcuts, and get to the correct answer in an efficient way.

### Why are mental math strategies important?

Mental math strategies are the foundations for most of the areas of mathematics that use numbers. Without efficient mental strategies, children can often struggle to quickly and fluently calculate.

Mental strategies are also the foundation of any written or formal method in mathematics. Referring to it as mental math does not mean you cannot write anything down at all, but any written work would be quick jottings to help remember through multi-step problems.

As children begin to use more formal methods, from around 3rd grade onwards, and as the numbers they are working with increase in value, mental math skills are vital for ensuring fluency and accuracy in math.

### Developing true fluency in mathematics

#### Effective mental strategies are important if children are to develop ‘true’ fluency

True fluency can be best defined as children being able to confidently use and apply their knowledge of number relationships, number facts and our number system in order to calculate and solve problems.

It is worth remembering that fluency in math is not simply restricted to being able to recall known facts. More accurately, it is how children can use and apply these facts, including through a range of mental math strategies, that are important.

“Low achievers are often low achievers not because they know less, but because they don’t use numbers flexibly.” – Jo Boaler

Third Space Learning’s one-to-one online tutoring focuses heavily on building pupils’ confidence and fluency in math. Tailored to each individual child’s needs, our weekly tutoring lessons aim to strengthen pupils’ understanding of number facts and how to apply them across a broad range of questions.

#### Be careful not to mislabel mental math skills

One important thing to remember when working to develop ‘true’ fluency is that accuracy is not the same as fluency.

For example, consider the following scenarios, which while accurate, may not necessarily be classed as fluent:

• A 1st grader calculating 40 + 8 by counting in ones;
• A 3rd grader calculating 1003 – 998 using a formal written method;
• A 5th grader calculates 41.79 + 25.3 + 25.7 – 41.79 by adding the first three numbers and then subtracting the fourth.

This extract from the research paper, ‘Developing Computational Fluency with Whole Numbers’ published in 2000 by S J Russell, remains one of the best explanations of fluency:

‘Fluency rests on a well-built mathematical foundation with three parts:

• an understanding of the meaning of the operations and their relationships to each other — for example, the inverse relationship between multiplication and division;
• the knowledge of a large repertoire of number relationships, including the addition and multiplication “facts” as well as other relationships, such as how 4 × 5 is related to 4 × 50;
• and a thorough understanding of the base ten number system, how numbers are structured in this system, and how the place value system of numbers behaves in different operations — for example, that 24 + 10 = 34 or 24 × 10 = 240’.

### Rapid recall vs mental calculations with notes

When we discuss mental calculations in math at upper elementary school, we need to be clear about the distinction between facts that children should be able to rapidly recall vs the types of calculations that children should be able to calculate mentally, sometimes with the support of notes.

Retrieval practice and rapid recall of number facts is important because if children are able to recall number facts automatically, it allows them to free up their working memory when faced with more complex questions.

They are also able to more efficiently and accurately solve problems, reason and make connections if they are not having to repeatedly calculate the same ‘basic’ facts.

“In teaching procedural and factual knowledge, ensure the students get to automaticity. Explain to students that automaticity [with key number facts] is important because it frees their minds to think about concepts.” – Daniel Willingham – cognitive scientist, in ‘Is it true that some people just can’t do maths?’

### The concept must be understood before introducing the strategy

Before we can expect rapid recall and automaticity of number facts with our mental math strategies, we need to teach the underlying math concepts. For example, only when children have a secure conceptual understanding of number bonds to 10, should rapid recall be attempted.

From this understanding of number bonds to 10, the strategy of partitioning can be used. For example, by 5th grade most children should be able to calculate 34 x 5 mentally (30 x 5 + 4 x 5) using partitioning and their knowledge of the distributive law supported by basic workings.

Although students will be learning more and more math facts which they can recall ‘by rote’, it is vital that they understand the concepts. Working with manipulatives can help with this, moving to virtual manipulatives on the interactive whiteboard when the numbers get too big to physically hold.

### Assessing the mental math strategies that your class is using

One really interesting way to check mental math strategies is to present groups with different written versions of the same math problems.

1. Present a single problem

If you present a problem, such as 64 + 17, in a sentence such as this, those children who are confident in their mental math strategies, will work it out in their heads.

They will usually, even subconsciously if they are fluent, partition the numbers and work out 60 + 10 and then 4 + 7, or 60 + 17 then add the 4. Some will do 64 + 10 and then add the 7.

Some may round the numbers, so say 64 + 20 using their number bonds to 20 knowledge and then minus 3.

Some may again use their numbers bonds to work out 64 + 17 by adding 63 + 17 to make 80 and then add 1.

You would expect your class to give a range of answers regarding their method, but hopefully all are fluent and can find the correct answer without any more than a quick jot down of some numbers if adding multiple steps.

1. Create two versions of the same set of 10 questions

Now put together a sheet of 10 similar questions, with a range of addition and subtraction which you would expect your class to be able to do mentally. Create a second version of this which lays out the same questions, with the same exact numbers and same expected answer, in standard algorithm method format.

Give half the class the first sheet laid out as a number sentence and the other half the second version where the questions are laid out in a standard algorithm method format.

Do not tell groups that they have different sheets and hand them out to different tables so they do not see the other format of the same questions. Give them time to individually complete the questions and write down their answers.

1. Ask children to share their methods

Take the first question and ask someone to volunteer to share their method. Then ask someone else to share, then someone else, and so on. Ensure you get a couple of examples from tables who have the horizontal layout of questions, and a couple of examples from the tables who have the vertical column layout.

You will likely find that the groups who had a horizontal layout were much more likely to have just worked it out mentally, whereas the groups given the vertical layout will have spent time doing the standard algorithm method to find and write down their answers, including every step, even though they could have easily completed those problems mentally.

This activity is a great reminder that even when we see a formal calculation we should be using our mental math strategies to speed up where we can.

### Building confidence in mental math strategies

When any new math concept is introduced – from addition through to percentages and decimals – children will benefit from being shown a physical representation of the numbers (using math manipulatives) and operations before using pictorial representations (such as number lines or bar models) and then finally written methods using the symbols of number and operation.

Read more: Concrete Pictorial Abstract Method

Along the way there will need to be lots of repetition and practice recalling facts mentally. As children get older, into upper elementary, the move from physical to written will hopefully get quicker for new concepts as they are building on a solid foundation.

Different children may be able to move to mental strategies at varying points of each unit. Some may jump from physical to mental if they grasp the concept quickly and have a sound understanding already.

Others may not be able to reach fluency of recall and application until they have had a lot of practice with writing their answers down and building confidence in those new number facts and strategies.

You may also need to unpack any misconceptions through those stages too, and this may involve going ‘backwards’ to the physical. It is good practice to always have manipulatives available during independent tasks, even in 5th grade and for all abilities. Sometimes a quick comparison using some Base 10 or Cuisenaire rods can help a child to ‘fix’ that strategy in their head.

It is also important not to teach children to do ‘tricks’ in math such as “add a zero” when multiplying by ten as this can cause issues in later years with their understanding of place value. You would hope, however, that children spot such patterns in their answers and this should lead to discussion and comparison as well as presenting opportunities for children to test their theory, where they have spotted a possible pattern. Even if you know it is wrong/right, they will gain from the chance to test and apply that assumption.

Read more: Math tricks to avoid

### The mental math strategies children should know by the end of 5th grade

By upper elementary there are some specific mental math calculations that will help children immeasurably when working in both written work and mental work, arithmetic and reasoning. They actually form a progression starting from 3rd grade, so it is important that the groundwork has already been done in Kindergarten – 2nd grade to enable children to carry out the calculations mentally.

These skills are therefore best looked at as a progression, rather than a set of year group expectations.

#### How to improve mental math year by year

As well as building on children’s range of mental calculations as they progress through elementary, make sure they are also secure in their number facts each year.

### How to develop the mental math strategies needed for addition and subtraction

In lower elementary, children will learn basic number facts including addition and subtraction. This will include number bonds to 20 by the time they finish 1st grade. They will do a lot of work with physical objects and role play, so it is good practice in these years to not just practice math skills during the math lessons, but also to make opportunities for questioning outside of these lessons.

Ask children to count up how many students are absent today, counting the pencils on each table to see if they have enough (or too many or too few) and reinforce vocabulary from math lessons.

Once children have grasped the concept of addition facts and subtraction facts, and that they are inverse operations (they may not know that specific word yet though) they will begin to solidify their rapid recall of number bonds and apply them to their work.

It is never too early to introduce different strategies to work out their calculations either, so long as the base understanding is correct. Asking them if there is another way they could have found the answer is a question which can be asked in formal lessons, in role play or in sports.

### Mental addition strategies and mental subtraction strategies at upper elementary

#### Counting forward and backwards

Counting forwards and backwards is first encountered in lower elementary, beginning at one and counting on in ones.

Students’ sense of number is extended by beginning at different numbers and counting forwards and backwards in steps, not only of ones, but also of twos, fives, tens, hundreds, tenths and so on.

#### Progression in counting forwards and backwards

1. Counting on or back in tens from any number (e.g. working out 27 + 60= ? by counting on in tens from 27)
2. Counting on or back in fives from any multiple of 5 (e.g. 35+15=? by counting on in steps of 5 from 35.)
3. Counting on or back in hundreds from any number (e.g. 570 + 300= ? by counting on hundreds from 570.)
4. Counting on or back in tenths and/or hundredths (e.g. 3.2 + 0.6 = ? by counting on in tenths. 1.7 + 0.55=? by counting on in tenths and hundredths.)

#### Partitioning for addition and subtraction

Partitioning strategies teach children how to break up larger numbers into smaller ones.

It is important that children are aware that numbers can be partitioned – both along the place value boundaries (canonically) and in other ways (non-canonically).

They can then use their partitioning to help them calculate addition and subtraction calculations. This can be extended as children progress through upper elementary.

#### Progression in partitioning

1. Calculations with whole numbers which do not involve crossing place value boundaries. E.g. 23 + 45= ? by 40 + 5 +20 + 3 or 40 + 23 + 5
2. Calculations with whole numbers which involves crossing place value boundaries. E.g. 49 – 32= ? by 49 – 9 – 23 or 57 + 34 = ? by 57 + 3 + 31
3. Calculations with decimal numbers which do not involve crossing place value boundaries 5.6 + 3.7= ? by 5.6 + 3 +0.7  or 540 + 380= ? by 540 + 300 + 80 or 540 + 360 + 20
4. Calculations with decimal numbers which involve crossing place value boundaries. E.g. 1.4 + 1.7= ? by 1.4 + 0.6 + 1.1 and 0.8 + 0.35 = ? by 0.8 + 0.2 + 0.15

Compensation involves adding more than you need and then subtracting the extra.

This strategy is useful for adding numbers that are close to a multiple of 10, such as numbers that end in 1 or 2, or 8 or 9.

The number to be added is rounded to a multiple of 10 plus or minus a small number.

For example, adding 9 is carried out by adding 10, then subtracting 1. A similar strategy works for adding decimals that are close to whole numbers.

1. Compensating and adjusting to 10. (e.g. 34 + 9=? by 34 + 10 – 1 or 34 – 11= ? by 34 – 100 – 1 = ?)
2. Compensating and adjusting multiples of 10. (e.g. 38 + 68= ? by 38 + 70 – 2 or 45 – 29 = 45 – 30 + 1)
3. Compensating and adjusting multiples of 10 or 100. (e.g. 138 + 69= ? by 138 + 70 – 1 or 299 – 48 = 300 – 48 – 1)
4. Compensating and adjusting multiples with decimals. (e.g 2 ½ + 1 ¾ by 2½ + 2 – ¼  or 5.7 + 3.9 by 5.7 + 4.0 – 0.1)

#### Calculating using near doubles

When children have an automatic recall of basic double facts, they can use this information when adding two numbers that are very close to each other.

1. Near doubles to numbers under 20. E.g. 18 + 16 is double 18 and subtract 2 or double 16 and add 2.
2. Near doubles to multiples of 10. E.g. 60 + 70 is double 60 and add 10 or double 70 and subtract 10 or 75 + 76 is double 76 and subtract 1 or double 75 and add 1.
3. Decimal near doubles to whole numbers. E.g. 2.5 + 2.6 is double 2.5 add 0.1 or double 2.6 subtract 0.1.

### How to develop the mental math strategies needed for multiplication and division

As students move through elementary school they will learn multiplication facts. They will need fluency in multiplication facts to enable them to recall these fast enough for testing now and in higher education. It is again vital that they understand the concept of multiplication rather than simply parroting the facts, rote style.

Though, practice is crucial as daily recall of known facts is vital to stop new facts pushing out the old where they are not fully embedded.

Children start multiplication understanding with doubling and halving in early elementary. This introduces the concepts of both multiplication and division and they should start noticing the patterns of these and apply this to math questions.

They will also learn multiplication facts for 5 and 10 and this starts with counting forwards and backwards in 5s and 10s, which they should also be doing from any given number not just zero.

By the end of 3rd grade, students should be able to recall all products of two one-digit numbers by memory. And then in 4th grade the 11 and 12 multiplication facts. They should also be applying these to word problems and multi-step problems as confidence increases, to ensure they are able to apply number facts rather than simply repeat them.

These mental arithmetic skills, and the fluency of them, will be vital in test situations. By the time they go to high school, they should have a very firm grasp of the number system along with known facts and patterns.

#### Place value multiplication strategies

Children should be able to build upon their rapid recall of 1-12 x multiplication and division facts, and multiplication and division facts for multiples of 10 and 100 to calculate an increasing range of multiplication questions mentally.

1. Multiply a 2-digit number by a single-digit number by partitioning. E.g. 26 x 3 = 20 x 3 + 6 x 3
2. Multiply a decimal number with up to 2 decimal places by a single digit by partitioning. E.g. 3.42 x 4 = 3 x 4 + 0.4 x 4 + 0.02 x 4

#### Doubling and halving strategies

Children should be able to recognize halving as the inverse of doubling and be able to rapidly calculate doubles and halves of numbers.

Some double and half facts are rapid recall rather than ones that children should need to calculate each time, and these are covered in the lists above.

1. Find the doubles and halves of any two-digit number and any multiple of 10 or 100. (e.g. half 680 or double 73)
2. Multiply and divide by 4 by doubling/halving twice and 8 by doubling/halving again. (e.g. 34 x 4 = 34 x 2 x 2.)
3. Find the doubles and halves of any number up to 10,000 by partitioning. (e.g. half of 32,202 by halving 3,000, 2000, 200 and 2.)
4. Multiply by 50 by multiplying by 100 and halving. (e.g. 8 x 50= 8 x 100 divided by 2)
5. Divide a multiple of 25 by 25 dividing by 100 then multiplying by 4 (by doubling and doubling again). (e.g. 350 ÷ 25 = 350 ÷ 100 x 2 x 2)
6. Divide a multiple of 50 by 50 by dividing by 100 then doubling. (e.g. 450 ÷ 50= 450 ÷ 10 x 2)
7. Double and half decimal number with up to one decimal place by portioning. (e.g. half of 8.4 by halving 8 and halving 0.4)

### Mental calculation strategies for fractions, decimals and percentages

As they progress through elementary school, children should develop their understanding of fractions, decimals and percentages and how they are related to division.

By 5th grade, they should therefore be able to use their rapid recall of multiplication and division facts to calculate some questions involving fractions, decimals and percentages, mentally.

1. Mentally find fractions of numbers in the 2,3,4,5 and 10 times table using known multiplication and division facts. (e.g. 3/5 of 45 by 45 ÷ 5 x 3.)
2. Recall percentage equivalents to ½, 1/3, ⅕, ⅙, 1/10 and 1/100. (e.g. ¼ = 25%)
3. Find 10% or multiples of 10% of whole numbers and quantities. (e.g. 30% of 50 by 50 ÷ 10 x 3)
4. Mentally find 50% by halving and 25% by dividing by 4 or 2 of numbers and quantities. (e.g. 25% of 150 by 150 ÷ 4)

#### Mental math percentages hack

The tweet below is something that you may have seen going around Twitter in early 2019, but it represents a useful strategy to help work out tricky percentages. This viral tweet is a very good example of a mental math trick. It’s also a great way to use mental math tricks to impress your friends!

### Top Mental Math Tips: How You Can Teach Mental Math Strategies

We’ve dealt with the ‘what’ in significant detail, but how do we actually go about teaching mental math strategies? Here is a summary of our top tips:

1. Teach mental math strategies and mental calculation techniques, don’t just rely on children ‘picking them up’. It is important that lesson time is devoted to teaching strategies conceptually and supporting children to make connections between their known facts and mental calculations. This is best achieved through modeling and the use of manipulatives etc.
2. Engage children in discussion. Children should be encouraged to discuss their mental strategies with each other and as a class, and adults in the classroom should join in this discussion. Children will see and approach calculations mentally in different (and equally as valid) ways and through sharing these, they expose each other to different ways of thinking about and ‘seeing’ a calculation.
3. Provide regular mental math practice. Children should have regular mental math practice that focuses on mental calculation strategies. Alongside teaching the strategies in the main math lesson, schools where children have a high level of competency and fluency in mental strategies, often devote 15-20 minutes a day to the practice and development of mental strategies and rapid recall outside of the main math lesson.
4. Don’t think that timed testing is the only way to achieve rapid recall. Timed testing has been shown by many research studies to be one of the least effective ways of developing rapid recall. Instead, ensure children have plenty of opportunities to use, apply and recall the facts that you want them to be able to recall rapidly.
5. Play games and create opportunities for meaningful activities. If the activities are fun and meaningful children will be supported in developing number sense and fluency in an increasing range of calculations.
6. Ensure ‘basic’ number facts are practiced. It’s important that you do not neglect ‘basic’ number facts, for example, number bonds within 10, 20 and 100 and the 1-12x multiplication table. Often facts such as number bonds are only practiced at lower elementary, but it is vital that these are practiced and children are encouraged to use these facts in their mental calculations. Remember, if you don’t provide the opportunity for them to use it, they will lose it!

References:

Russell, Susan Jo (2007). Developing  Computational Fluency with Whole Numbers in the Elementary Grades

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The content in this article was originally written by math consultant and author Tim Handley and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza. ##### Tim Handley
Third Space Maths Consultant
Author
Tim Handley is a maths consultant and author, working as part of the Third Space Learning team to create resources and blog posts.
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