How to Teach Two Digit Addition Without Regrouping

Second grade is a very important year where students develop fluency with two-digit addition and subtraction. It is the year that we work on a multitude of addition and subtraction strategies that students can use to solve problems. We spend a lot of time discussing a variety of strategies, using many different models, and doing mental math.

Why? To develop students' flexibility when solving math problems .

The Common Core Standard for two-digit addition & subtraction is:

CCSS.MATH.CONTENT.2.NBT.B.5
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

And, the standard for three-digit addition and subtraction, to show where we're headed:

CCSS.MATH.CONTENT.2.NBT.B.7
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Nowhere in those two standards does it say anything about the standard algorithm that we all learned in school (most likely with the language of "carry" and "borrow"), nor is the standard algorithm directly addressed in the Second Grade Common Core Standards. Read to the end to find out how I address the standard algorithm in our classroom.

Are you interested in a free sampler of some of my Two-Digit Addition and Subtraction Products?

Strategies vs. Models

If you are familiar with my Addition & Subtraction Word Problems, you may have noticed that I make a big distinction between the strategies used when solving problems and the models students employ with those strategies.

Strategies are usually how students approach and manipulate the numbers. Models are how the strategies are organized on paper so that students can explain or see the strategy.

When looking at the standards above, I can see that the strategies are clearly noted in the standard:

In 2.NBT.B.5 and the strategies are:

• place value
• properties of operations
• relationship between addition and subtraction

Standard 2.NBT.B.7 even notes that the models or drawings (which I also call models) are separate from the strategies that are based on:

• place value
• properties of operations
• relationship between addition and subtraction

As you can see, the strategies are clearly outlined in the standards. Now within each of the above general strategy categories, there really are many different strategies that students can use and you can label them whatever you'd like in your classroom. I like to label them with students' names as an easy reference. That way, we can refer to Samantha's strategy when solving a problem. Or you can label the strategy with the action that the student takes in the problem (for example: Add Tens First).

However, I still make a distinction between the strategy and the model. Why? Because students can use multiple strategies with one model. There's no one right way to use the model, as long as the student can explain his or her thinking. The models (or drawings) merely give students a tool to explain their thinking on paper or with manipulatives. The thinking, or what students do with the numbers, is the strategy. What they use to show it to you is the model.

In all honesty, I'm not always consistent in labeling something a strategy or a model. I try to be, but like you, I'm human and sometimes mix them up, especially when I'm in the moment with students. It's a learning process and something I'm continually reflecting on throughout the years. All that to say, you may see a few things labeled one way and question its label. Go ahead and question it, think about it, mull it over and figure out whether it's accurate or not. All of this is still new to many of us.

Here are some anchor charts that I've used the past couple years that illustrate some of the below models and strategies.

Models for Two-Digit Addition

Below are a few models that we use do two-digit addition or subtraction. Are these the only models you can use? No, this is not an exhaustive list.  They are what I have found useful in the classroom for students to practice and use to build conceptual understanding and number sense.

Number Lines for Two-digit addition and Subtraction

I usually start with number lines when I introduce students to paper / pencil models. An open number line is very flexible. Students can make jumps of one or ten (or more) and easily manipulate it to show their mathematical thinking.

I usually help students get to the nearest 10, or friendly or benchmark number when using a number line because it is easier to make jumps of 10. That is an example of the difference between a model and a strategy.  The model is the number line.  The strategy is making jumps of 10.

Teaching how to use number lines when using 10 to add +9 and +8 facts, solidifies this strategy when students are adding larger two-digit numbers.

Remember, the number line is the model and it can be used with a variety of strategies. Modeling and practicing using a number line with easier problems will help students when using a number line with more difficult problems.

One of the daily activities that we do with numbers lines is our Daily Math. This is a whiteboard sheet that we go through daily. The number line at the bottom helps students solidify their understanding of both how to use a number line and how "make 100 or make 1000".

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  Daily Math – Number Sense, Addition and Subtraction, Time and Money

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Here are a few more examples of how we use number lines in the classroom.

This is from my Roll & Spin Math Stations.  In this activity, students practice making jumps of 10 and 100 up a number line.

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  Roll and Spin Math Games

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There are also versions where students subtract 10 and 100 down a number line, too.  One of the skills students need to be successful on number lines is the ability to make jumps of 10 and 100.

This is an example from one of our Addition & Subtraction Word Problems where students had to figure out a separate start unknown problem.  This student started at 15 and counted 35 jumps and then took one away at the end.  This is also a great example of compensation (see below) because the student added one to the 34 to make easier jumps and then took it away at the end.

This is from my Second Grade Cut & Paste Math Activities.  In this activity, students are practicing how to add up, starting at the smallest number and figuring out who tot get to the larger number by jumping to the friendly numbers.  This student started at 19, jumped to 20, then made jumps of 10 to 60 and made a jump of 3.  The student added their jumps together to get 44.

The above are a few examples from my Two-Digit Addition Math Stations.  My students needed more direct practice with number lines and making jumps, despite all of our whole group practice.  So, I gave them the directions and students followed them on the number lines.

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  Two-Digit Addition Task Cards, Assessments, Activities, and Games

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A more recent resource that I developed to help students develop number fluency is the Make 100 and Make 1000 resource. This resource has MANY activities where students practice making 100 and making 1000. Number lines are one of the activities.

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  Make 100 and Make 1000 Activities and Printables

  $5.25

I also have a whole blog post on how to use a number line with even more examples of how to develop number line fluency in the classroom.

Base-10 Blocks

Base-10 blocks are another model I teach students to use; however, I generally teach students to draw the base-10 blocks. We do use real foam blocks in class, but I try to move away from them as quickly as possible.

Why? Students will always have pencil and paper to solve problems, but they won't always have manipulative available to them. Using base-10 blocks also takes a lot of time. I don't mind spending the time on them, for students who need them, but I also want to push students toward more efficient tools.

Here are a few examples of how we use base-10 blocks:

The above two are using base-10 blocks by drawing out the tens as "sticks" as we refer to them in our classroom.  These particular students were having difficult counting over 100 by tens, so I had them draw each number in tens, then count by tens until they got to 100, then start over counting by 10s again.  Not only did this help them add up numbers beyond 100, but it also gave them more expense with our base-10 number system.

The above example is from my Two-Digit Addition Math Stations again and is just a basic problem – answer matching with base-10 block representations.

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  Two-Digit Subtraction Assessments, Task Cards, Activities, and Games

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The Number Line blog post also has an interesting visual activity to help students transition from base-10 blocks to number lines.

Strategies for Two-Digit Addition

As noted above, the main three strategies stated in the standards are:

• place value
• properties of operations
• relationship between addition and subtraction

Below are a few strategies that we use to solve two-digit addition problems. Most of them are based on place value strategies as I find those tend to be easier for students to understand and apply. Again, these are how students manipulate the numbers in the problem to make it easier to solve.

No one strategy is the "right" strategy for every student for every problem. Some problems lend themselves to certain strategies because of the numbers. Students may also switch between strategies within the same problem, depending on how they're manipulating the numbers. The key thing to look for is if the student can explain his or her thinking when solving a problem.

Break Apart or Ungroup (Place Value)

This strategy requires a bit more mental math practice, but it can be so powerful. The basic idea is that the number is broken apart into tens and ones and then, either using a number line, base-10 blocks or just numbers, students manipulate the pieces to add or subtract the numbers.

Breaking the number part or ungrouping it helps students see the value of place value. The tens place is not just 4. Its value is 40 or 4 tens.

One resource that helps develop this strategy is the Number Talks book (affiliate link).  We do number talks through the year, starting with addition facts and moving into two-digit addition and subtraction by the end of the year. I love seeing the strategies that my students can come up! The Number Talk book is also a great book that helps develop listening skills.

Think about the problem 64-47. Students break apart the problem into 50+14-7-40 and take away the parts by place value. I'd probably start with the 14-7, but students could start anywhere that makes sense for them.

The above examples come from my Two-Digit Addition Math Stations and illustrate how students can break apart numbers and add up each place value.  Breaking apart is also called ungrouping or decomposing, depending on the math program you use.

Did you notice that in one of the problems above, the student added 60 +40 and got 106, yet he wrote the correct answer to the problem?  What do you think was going on with this student?  Do you then he couldn't add 60+40, made a silly mistake, or is there another reason he wrote the 106?  Seeing students interact with these types of strategies will give you a place start conversations with them about their mathematical thinking.

One more example from some Addition Task Cards where students only break apart the second number then make jumps of 10 and 1 using 100s and 1000s charts.  Although we give plenty of practice using a 100s chart in first grade, I find that students don't necessarily transfer their learning to larger numbers in second grade.

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  Addition Task Cards Using 100s Charts

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Add Tens to Tens and Ones to Ones (Place Value)

This is very similar to the break part strategies, except without breaking apart the numbers. Students can add the parts of the number (the tens or the ones) together mentally because they know their addition facts. We basically use a v-model to draw lines connecting the tens and adding or subtracting those parts.

Here is one example of how we've used it in the classroom:

Subtract Tens, Subtract Ones (Place Value)

Similar to add tens to tens and ones to ones, students subtract each place value separately and then subtraction the ones from the tens (or add it). There are basically two ways to use this strategy. Students can decompose a the ten or students can use negative numbers.

One way that I use this strategy with students is with negative numbers. I know we don't teach negative numbers in second grade, but for some students, this is really a way that they understand and can hold onto more than the other strategies.  You can see examples of this in the second and third anchor charts above.

Think about 64-47. If I subtract 4-7, I get -3. I tell students that the bigger number has the minus sign in front of it and so it still has more that needs to be taken away. Students then subtract 60-40, get 20 and subtract there more to get 17.

Count Down / Think Addition (Count Up) / Add Up (Relationship between Addition & Subtraction or Place Value)

I'm not exactly sure whether this strategy is about the relationship between addition and subtraction or place value. The Think Addition Strategy is similar (if not the same as) Count Up or Add Up. This strategy is also very similar to the Break Apart Strategy, in that students need to break at least one of the numbers apart to sound up or down by the parts of the number.

Although students can count by ones, I highly encourage you to help them move toward more efficient strategies and count by tens and then ones. Using a hundreds chart gives students practice moving by 10s up and down the chart. A hundreds chart is sort of like a compressed number line.  See the above photo with the 100s and 1000s charts.

Here are a few examples of counting up:

The above two examples are just ones we did on the whiteboard and I had students write down in their notebooks.

This is a page from my Two-Digit Subtraction Flap Books.  These Flap Books go through several different models and strategies and give students practice with vocabulary and explaining their thinking.

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  Two-Digit Subtraction Flap Books

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The thing I LOVE about these flap books are that students can dive deep into one aspect of two-digit subtraction and attach language to the numbers and processes that they use.

Use Compensation (Properties of Operations)

This last strategy is unlike any of the previous ones. It basically has you make sure that the numbers are balanced within the problem and that you're accounting for all of the parts. It's a precursor to algebra and a great strategy for mental math.

There are a couple different ways to use compensation, but the basic idea is that you add or subtract some of one number and add it to the other number to create a friendly number. You have to keep track of what was added or taken away and account for it somehow in the problem.

Compensation is especially useful for numbers that are close to friendly numbers, although it can be used for any number. For example, 68 – 39 could be transformed into 69 – 40. I've added one to each number. The value of a +1 and -1 is 0, so I haven't changed the problem at all.

Here's another example: 53 + 38. I might add 53 + 40 and get 93, but because I added two to the 38 to get to 40, I'll need to subtract two from 93 to get 91.

The basic idea with compensation is that you are adjusting one part of the number into a friendly number to make it easier to add or subtract. However, when you adjust one number, you have to keep track of what you've adjusted and compensate for it.

What do students need to know before using these strategies?

The above strategies are very powerful if students can add them to their toolkit when approaching two-digit addition and subtraction. However, to effectively use the above strategies, students need a few things in place.

Addition and Subtraction Facts – Students need pretty good fluency with their addition and subtraction facts. Do they need to have all of them memorized with speed? No. However, if students are spending too much time trying to figure out an addition fact and it's keeping them from focusing on the strategy because they forget what they were doing, then they need more fluency with their addition and subtraction facts.  My Automaticity Assessments help students practice their facts by strategy.

Ability to find friendly numbers – At the beginning of the year, we spend a long time developing fluency with 10 as a benchmark number. Although we do it at the beginning of the year to help with our math fact fluency, it is also beneficial when students begin their journey with adding and subtracting two-digit numbers. Students need to know how to get to the next friendly number, which is essentially their 10s facts but applying them to two-digit numbers to find the next ten.

Adding 10 to a number – We start our two-digit addition unit with a lot of practice adding and subtracting ten from a number. This is a foundation skill in both my two-digit addition products as well as my two-digit subtraction products. Students must see the pattern of adding 10 to a number.

Place Value – To do two-digit addition, students need a strong foundation in the concept of ones and tens and what it means to break a number apart into ones and tens. From the first day of school, we are doing Daily Math exercises that build fluency with place value as well as skip counting by 10s from any number.

Do I teach the traditional algorithm?

Yes and no. Yes, I teach the concept of regrouping and yes, I do teach students to move toward efficiency when adding and subtracting. That could include the traditional algorithm if they can understand the meaning behind it.

Students do not need to use the standard algorithm until fourth grade (according to the Common Core Standards). Can they do it earlier?  Maybe.

I expose them to it in second grade as a model they could use; however, we don't spend a lot of time focused on it, because I want students to develop strategies for solving problems, not be tied to one model.

When we do work with the traditional algorithm, we attach a lot of language and meaning to it, generally tying it to work we've already done, like our work with base-10 blocks. Here are a few examples of I teach students the traditional algorithm by linking it to models we've already used and giving students accurate language to use to explain their thinking.

Here are a few examples of how I give students experience with the traditional algorithm.

Did you notice that should say 7 tens and 11 ones?  The student didn't pay attention to the base-10 blocks!

These come from my Decompose a Ten packet, which balances work the traditional algorithm with base-10 models and gives students the language of decomposing numbers.

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  Decompose a Ten

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Whew – that's a lot of information to digest! There's many different models and strategies a student can use to solve two-digit addition and subtraction problems. What I outlined above are a few that I have found especially helpful for students. They help students develop a solid foundation with two-digit addition and subtraction, create a bridge to three-digit addition and subtraction, as well as emphasize the idea of using strategies and models to solve problems, not just following steps in a process.

If you teach second grade, you might like a few pages from some of my two-digit addition and subtraction products. I've compiled this PDF of resources as a sampler from several different products that really emphasize all the work we do in our classroom to develop these strategies in depth.

Different components of the sampler can be used whole group or small group and are perfect for helping your students think outside the box when it comes to solving multi-digit addition and subtraction.

Two-Digit Resources Mentioned Above

Here is a list with links of all of the two-digit addition and subtraction resources mentioned above.  They can be purchased on my website or on Teachers Pay Teachers.

• Roll and Spin Math Stations
• Cut and Paste Math Activities for Second Grade (TpT)
• Two-Digit Addition Math Centers (TpT)
• Two-Digit Subtraction Math Centers (TpT)
• Addition Task Cards Using 100s Charts (TpT)
• Two-Digit Subtraction Flap Books (TpT)
• Decompose a Ten Task Cards (TpT)

Many of the above are also included in a Two-Digit Addition and Subtraction BUNDLE (TpT).

Additional Two-Digit Addition & Subtraction Resources

• Two-Digit Addition Number Puzzles (TpT)
• Two-Digit Subtraction Number Puzzles (TpT)
• Decompose Two-Digit Numbers Number Puzzles (TpT)
• Two-Digit Addition No Prep Printables / Worksheets (TpT)
• Two-Digit Subtraction No Prep Printables / Worksheets

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  Roll and Spin Math Games

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• 

  Addition Task Cards Using 100s Charts

  $4.79

• 

  Decompose a Ten

  $3.75

• 

  Two-Digit Subtraction Flap Books

  $5.39

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  Two-Digit Subtraction Assessments, Task Cards, Activities, and Games

  $9.57

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  Two-Digit Addition Task Cards, Assessments, Activities, and Games

  $9.57

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  Two-Digit Addition No-Prep Printable Practice

  $9.57

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  Two-Digit Addition and Subtraction BUNDLE

  $48.57

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  Cut and Paste Math Activities for Second Grade – Numbers and Base Ten

  $5.75

How to Teach Two Digit Addition Without Regrouping

Source: https://www.whatihavelearnedteaching.com/models-strategies-for-two-digit-addition-subtraction/

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