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# Teaching Addition and Subtraction: A Guide For Elementary School Teachers From 2nd To 5th Grade

Addition and subtraction in upper elementary school math builds on the foundational skills students acquired in lower elementary school, helping them perform more sophisticated mathematics, and solve more complex problems.

This post will show you what that progression looks like from 2nd grade to 5th grade, what your students should be able to do before moving on, and finally offer some practical suggestions as to how some objectives could be taught in the classroom.

#### What is addition and subtraction?

Addition and subtraction are two of the ‘four operations’–the four core math concepts children need a strong understanding of in order to tackle the rest of the subject. Addition is the act of putting two or more numbers together to obtain a larger result, and subtraction is the reverse – removing one or more numbers from another to obtain a smaller result.

Addition and subtraction are among the first math skills children are taught, and are key in developing their number sense.

Earlier in elementary school, students learn basic addition and subtraction. Kindergarten students should be able to solve addition and subtraction problems within 10 and develop methods to compose and decompose numbers within 19.

1st grade students should be able to solve addition and subtraction problems within 20 (i.e. one and two-digit addition) and develop methods to add within 100 and subtract multiples of 10.

Students will only encounter relatively basic number sentences, are not required to write their own number sentences (however, this should be encouraged) and most teaching will incorporate manipulatives and simple visual representations, such as flashcards and number lines.

It is an understatement to say that a secure understanding of how place value works in base 10 is a key component to success in addition and subtraction in math.

Along with students developing their mental models of number and understanding the ‘numberness’ of numbers between 1-20 e.g. four can be made the following ways:

• 4+0
• 1+3
• 2+2
• 2+1+1
• 1+1+1+1

Number bonds within 20 are also a key element that should be near to a high degree of fluency – meaning that students should be able to solve these problems mentally – by this stage.

Students that have this conceptual understanding of numbers and the number system are far more likely to be successful when it comes to manipulating numbers in elementary math and beyond, especially when tasked with doing so mentally.

Therefore, before you begin teaching this unit, it is worth knowing that when we teach for mastery, our first step is to ensure the prerequisite knowledge needed to be successful – it is not enough to rely on the fact that students have progressed into 2nd grade that they fully grasped the fundamentals and are ready for 2nd grade content.

It is recommended that regardless of the curriculum your school follows, you ensure that you’ve finished teaching place value before you begin addition and subtraction in 2nd grade.

If you are comfortable that your students are secure, then carry on reading.

### Teaching addition and subtraction: The theory

It’s important to remember that students should still be using manipulatives at this point to help them with their conceptual understanding of the mathematical knowledge they are gaining.

A possible error that new teachers may fall into is that because curriculums, including the Common Core State Standards, mentions that calculations should be done ‘mentally’, they may take that to mean that manipulatives cannot be used.

Here it is worth reminding teachers that these objectives are the outcomes; it is totally appropriate (and I would argue necessary) to use manipulatives as part of a concrete, representational, abstract (CRA) approach when beginning to teach this unit.

Of course, the purpose of any manipulative is to show the underlying mathematical structure so that it is understood; then gradually reduce the need for its requirement. Math manipulatives teachers may want to consider to help teach this unit include:

• Place value counters
• Base 10 blocks

Visual representations a teacher should use, but eventually discourage use of include:

• Bar models
• Part-whole models
• Place value charts An example of use of part whole model from Third Space Learning’s online intervention tutoring platform

(Number lines are, by this point, too simple a representation to use with most students.)

If sourcing these manipulatives is hard, the Didax website has some virtual ones that can be shown on a large interactive whiteboard.

For states following the Common Core State Standards in 2nd grade, building fluency with addition and subtraction is one of the main focuses of instructional time. Below are some key skills expected of 2nd grade students.

• Add and subtract numbers, including:
• add up to four two-digit numbers
• Fluently add and subtract within 100
• add and subtract within 1000, using concrete models or drawings
• Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900
• Use addition and subtraction within 100 to solve one- and two-step word problems

For schools following the Texas state standards, or the TEKS, students will be expected to solve one-step and multi-step word problems, involving addition and subtraction, within 1,000 by the end of 2nd grade.

A good easy way into the objective ‘add and subtract numbers mentally…’ is to look at the addition and subtraction of multiples of one hundred. Base ten blocks are useful here as the students should be familiar with them from previous years.

Furthermore, you can demonstrate the relationship between ones, tens and hundreds by counting to 10 in ones, 100 in tens and 1,000 in hundreds. By asking students ‘what do you notice?’, they will soon see the relationship between the amount of physical pieces you have and the quantity they represent.

Once students have seen this pattern and are familiar, teachers can encourage whole-class skip counting, both forwards and backwards, playing games like showing a set amount of the 100 block, getting the students to close their eyes and hiding a set amount.

The students will then have to tell you how many there were originally, how many were taken and how many are left. This can then be modeled to show a formal calculation.

This will help improve the students’ understanding of place value and mental calculations. Furthermore, it will allow the students to feel successful, which will inspire them to try some of the more difficult objectives.

The next step would be to take multiples of 100 away from numbers that have a value in the hundred, tens and ones.

A typical example of a word problem that students may be expected to solve by the of this period of teaching would look like this:

There are 26 ducks in the pond. 45 more ducks arrive. How many birds are there?

For this addition word problem, students may use many different strategies to solve it. One student may take 4 from 45 and add it to 26 to make a more friendly number of 30. And then add 41+30 = 71. Another student may add the 10s first (20+40 = 60) and then add the 1s (6+5 = 11).

They would then add 60+11 = 71. It’s important to note that students using the formal algorithm method here is not the most important. Allowing students the ability to reason and think about how numbers are created can only help them in the long run.

Mr. Almond has 425 marbles in a box. He loses 39. How many are left over?

Similarly, the end goal of the unit would be for students to solve this subtraction word problem with the formal algorithm subtraction, but once again discussing possible mental subtraction strategies is highly recommended.

There is, of course, more to the learning of math than just learning these objectives, and reasoning and problem solving should not just be limited to word problems. These questions will help develop the reasoning and problem-solving questions from this unit.

Is this the most effective method? Discuss.

Once the formal algorithm method has been learned, there is a tendency to overuse it. By giving students questions like this, it reinforces that we are merely providing new mathematical tools for the learner to use as they wish when they deem it appropriate.

We need to be reminding students that there is always more than one method they can draw on.

Creating problems that vary slightly to what a student has typically seen or experienced, is a good way to see if a student understands the underlying math or is merely able to parrot a method back at you.

This bar model question may cause some issues at first as there are three numbers that have been added together to make the whole and the missing part comes between the two other parts and not at the end, as is typically seen in a classroom.

This problem allows greater discussion of the underlying mathematics particularly the commutative property of addition which would allow the students to rearrange the bars as below.

Read more: What is a bar model

It is these types of questions that will push students’ addition and subtraction skills and set them on the path to being true mathematicians. These skills can then be used in other topics – for example, as a key part of teaching statistics and data handling.

• Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
• Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity.
• Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

As well as revisiting the objectives from 2nd grade (remember those prerequisites), in 3rd grade students move towards fluently working with numbers within one thousand.

Because of the hierarchical nature of math (there is a certain order that knowledge of the domain needs to be taught in for the rest of it to make sense and stick), it is so crucial that students are comfortable with counting in 100s.

Building on the advice given in 2nd grade, considering using place value counters could be one such way into this unit – though it is hoped that by this point, students are secure in their understanding of place value.

For this section, I want to focus on the objective, ‘Assess the reasonableness of answers using mental computation and estimation strategies including rounding.’

Depending on the prior experiences of the students, using Cuisenaire rods can be helpful in showing this.

With plenty of practice of counting from the smaller number up to the larger number, coupled with the physical taking away using place value counters, students should quickly grasp this idea.

If students are already familiar with the idea they can move to the next part.

#### The standard algorithm: addition and subtraction

In 3rd grade, students encounter the formal written method of standard algorithm addition and subtraction. It is common practice for the standard algorithm in addition to be taught first, swiftly followed by the standard algorithm in subtraction.

This is an area that is generally taught well by teachers, as they would likely have been taught this method themselves when at school.

Assuming automaticity within both methods, when students reach 4th grade, it is possible to combine using the standard algorithm in  both addition and subtraction in order to ensure that students can use the inverse to check their answers.

As students are now becoming familiar and increasingly familiar with the algorithm method for addition and subtraction, now is an excellent time to add in one more step to the method, which is to perform the inverse calculation as part of the process. This would be what the students are familiar with already:

  4,532
+3,653
7,185


What I would propose is the following:

  4,532
+3,653
7,185
3,653 -
4,532


Here the students have taken the sum and an addend from the addition part of the calculation and used them to form a minuend and a subtrahend of a subtraction calculation.

The difference between the minuend and the subtrahend was also the first addend of the addition question. If this is the case, then the original question has been answered correctly.

A typical problem you would expect students to answer would be the following:

Mr. Almond buys a laptop for $482 and a tablet for$239. How much did he spend altogether?

Students should use a bar model to represent this addition word problem.

And then use a formal written method to solve – including the use of the inverse calculation at the end.

  482
+239
721
239 -
482

When looking at creating reasoning and problem-solving activities, it is highly appropriate to look back on the objectives from previous years and create a reasoning or problem-solving activity based on them.

In mathematics, maturation matters. If some ideas are still too novel for students, despite them showing some success with them, then solving problems with them can be overwhelming.

As we want students to attend to the mathematics, creating difficult problems with numbers they are more comfortable with frees up the students thinking to consider the structures and not worry about the numbers. A good problem to look at would be missing numbers in a calculation.

Plenty of reasoning is available in these questions. Students can look at the addend and sum and reason the missing number must be an even number as adding an even number to an odd number produces another odd number.

These questions also allow students to practice their fluency of number bonds. These questions can vary in difficulty by bridging to the next place value or not – something that students often find difficult during these tasks.

• Fluently add and subtract multi-digit whole numbers using the standard algorithm.
• Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
• Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

As students continue to develop their math skills in this area, it is hoped that they have developed a strong understanding of the place value system. In 4th grade place value, students learn numbers to at least 1,000,000.

It is therefore likely that the questions you use will include 4-digit numbers, and possibly up to 6-digit numbers.

To add an element of challenge into the teaching at this step, teachers could try some of the following:

1. Provide equations that require balancing on both the left and the right-hand side of the equal sign. E.g. 142,530 + 432,943 = 354,954 + ________
2. Ask students round the numbers in the question to a nearest whole place value to estimate the answer first.

E.g. 142,530 + 432,943 = 354,954 + 220,519

142,500 + 432,900 = 355,950 + 219,450

Doing this further enhances the students’ understanding of the equals sign while practicing rounding skills. You could challenge students to round the same numbers to the nearest hundred thousand, ten thousand, thousand, hundred and tens to investigate which rounding will give an estimate nearer the final answer.

At this stage, in accordance with the Common Core State Standards, students should be solving multi-step problems in context. A typical problem could be something similar to the following:

Journey                              Distance (miles)

London to Paris                       580 miles

London to Rome                     908 miles

Paris to Rome                         737 miles

A plane flies from London to Rome and then on to Paris.

How much further is this than flying direct to Paris from London?

908 miles + 737 miles = 1645 miles

1645 miles – 580 miles = 1065 miles

This question relies on students having an understanding of measurement and uses numbers that they are familiar with.

Again, with two-step problems, using slightly easier numbers is actually beneficial, as it allows students to concentrate on understanding the language and structure of the question as to why it is a multiple step problem. Students should encounter plenty of worked examples of these before attempting their own.

Reasoning and problem-solving in 4th grade gets more sophisticated. Numbers are exchanged for symbols and a greater use of unknown quantities is used to get students to reason about their knowledge of numbers, laying the foundations for algebra in high school.

A typical problem may take the form of the following:

The way to tackle a problem such as this is to ask students what is the same and what is different about the first two calculations. Teasing out that the difference between the two equations is one square and that the difference between the two answers is 700.

From this, students can deduce that a square is equal to 700 and therefore a triangle is 500.

When adding two triangles and a square together, you get the answer 1,700.

• Perform mental calculations, including with mixed operations and large numbers
• Use their knowledge of the order of operations to carry out calculations involving the 4 operations
• Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them
• Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
• Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

When looking at 5th grade word problems, pupils should be able to solve them in a range of real-world contexts (money and measurement, for example) as well as with up to 2 decimal places.

A common subtraction problem for this age group could be something on the lines of the following:

The Children and Green School are raising money for a charity. Their target is to collect $380. So far they have collected$77.73. How much more money do they need to reach their target?

The difficulty in this subtraction word problem is the requirement that when performing the subtraction, students must remember to include two-zero place holders from the target number to ensure the place value is correct.

380.00

77.73 –

Students will then have to cross three place value holders to the tens column in order to perform all the necessary exchanges, which adds an additional level of difficulty. Once the calculation has been performed, students should get the answer \$302.27.

The following question provides an insight into the type of reasoning and problem solving that is expected by the end of 5th grade.

Students are expected to take the information from the table and perform a 2-step calculation – find the combined height of Kilimanjaro and Ben Nevis and then the differences between this total height of both mountains and Everest.

This could be represented in the following way using bar modelling to make the two calculations clearer to see.

Adding both heights of Kilimanjaro and Everest gives a height of 7,239 meters. When this is subtracted from the height of Everest (8,848m), you are left with the answer 1,609 meters.

Addition and subtraction are the first mathematical skills that students really get to grips with, and having a strong foundation in them is key to becoming better mathematicians. Hopefully this post has given you some good ideas to achieve exactly that for your class, no matter which grade they are in!

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