The post Constant Difference: A Key Mental Math Subtraction Strategy For Quick Calculations appeared first on Shelley Gray.

If you find your head spinning when you think about all the addition strategies that you should be teaching, you are certainly not alone. Although teaching addition is one of the most important math concepts that we encounter, it comes with challenges.

First of all – time. How do we find the time to do a really good job of teaching the different addition strategies so that our students possess excellent understanding, while also doing a really good job of teaching everything else in our overwhelming curriculum?

Second, differentiation. All of our students learn at different speeds and in different ways. We can’t expect them all to learn the addition facts and strategies at the same time, but how do we ensure that each student is working to his full potential?

One last big challenge is the balance between mental math strategies and memorization. We know that strategies are important. We want our students to UNDERSTAND number, rather that simply memorizing the facts. However, automaticity is important too! How can we reach this balance?

Fortunately, there are specific strategies that we can teach to make addition easier for our students, and accessible for all of them.

Before I begin – if you are looking for a resource where all of the work is done for you, you may be interested in The Addition Station, a self-paced, student-centered math station where students work through the basic addition facts and strategies, mastering each one as they go. Strategies are integrated in a strategic manner, ensuring that students build on their understanding progressively. See The Addition Station for Grades 1-2 HERE and The Addition Station for Grades 3-4 HERE.

Alright, so let’s talk about addition strategies.

Strategies are ESSENTIAL, for all operations. We want our students to be able to think flexibly about numbers, and use strategies naturally. This means that understanding is key. Automaticity (quick recall) will naturally follow.

Here are some effective strategies for addition:

Plus 1, 2, 3 and Extensions

In younger grades, we begin with the Plus 1, 2, and 3 facts. We can teach Plus 1 as 1 more, Plus 2 as 2 more, and Plus 3 as 3 more. As our students are ready for more of a challenge, we can extend these facts into the tens, hundreds, and even thousands. For example, the fact 7+1 can be extended to 70+10, 700+100, or 7000+1000. Teach your students to look for familiar facts in these bigger problems, so that when they need to solve a fact like 50+20, they think, “I  know that 5+2=7, so 50+20=70.” During the extensions, be sure to emphasize place value. For example, we can think of 500+200 as 5 groups of 100 plus 2 groups of 100 to make 7 groups of 100.

Counting On

If students have been working with Plus 1, 2, and 3, they have technically already been working with the counting on strategy. Counting On is an introductory addition strategy that should only be used to add 1, 2, 3, or 4 to a number. Beyond this it gets confusing and can cause errors. To count on, we begin with the higher number and count on. For example, for 17+3, we think, “17…18, 19, 20.” For 2+34, we start with 34 and count on: “34…35, 36.” Dot patterns, ten frames, and number lines are all excellent tools for counting on. Read more about counting on, and download some free printables to help you HERE.

Extending the Doubles and Near Doubles

The doubles are typically facts that become automatic early on. The near doubles are facts like 4+5, where we encourage students to think, “I know that 4+4 is 8, and 1 more is 9.” We can also extend these facts. For example, when a student is faced with 30+30, he can think, “I know that 3+3 is 6, so 30+30 is 60.” Again in this level, we encourage students to think in groups of 10, 100, or 1000. For example, 200+200 can be thought of as 2 groups of 100 plus 2 groups of 100. Read more about extending the doubles and near doubles, and download some free printables to help you HERE.

Plus 7, 8, and 9

When we add 7, 8, or 9 to a number there are a couple of different approaches that we can use. First of all, we can add 10 and then take some away. Alternatively, we can make a 10 and then add the rest. I’ve discussed both of these approaches in detail in THIS BLOG POST. You’ll also find some free printables to help you!

Left-to-Right Addition

Left-to-right addition (also known as front-end addition or the partial sums method) is one of the most powerful mental math strategies for teaching addition of 2 or 3-digit numbers. However, many people are confused by why it is important and why it can be more effective than traditional vertical addition.

With left-to-right addition we add from left to right. So in a two-digit equation we add the tens first and then the ones. For example for 25+34 we first add 20+30 to make 50, then 5+4 to make 9, and then 50+9 to make the final sum of 59.

For a detailed explanation and rationalization of this strategy, as well as free printables to help you teach it, please see THIS POST.

Using Friendly Numbers

A friendly number is a number that is easy to work with. For example, multiples of 10 are “friendly” because they are easy to work with when we add or subtract.

When we use the “friendly number” strategy for addition, it helps us work with big numbers. This is because we are essentially breaking the equation up into more manageable parts.

We begin by getting to a friendly number, which is typically a multiple of 10, 100, or 100 – depending on the numbers that we are working with. Then we add on the remainder.

For example, for the equation 27+9, we could first get to the friendly number 30 by adding 3, and then add the remaining 6 to make 36.

For a complete, detailed explanation of the friendly number strategy for addition, as well as some free printables to help you, please see THIS POST.

Breaking Up An Addend

Breaking up, or decomposing, an addend is a fantastic mental math strategy for addition that can be used in many different circumstances. This strategy involves breaking up one of the numbers in an equation into more manageable parts. Like many other mental math strategies, this encourages students to think flexibly and to manipulate numbers in different ways. This is the big goal of mental math!

For a detailed explanation of this strategy as well as free printables to help you teach it, please see THIS POST.

Compensation

Compensation is a mental math strategy for multi-digit addition that involves adjusting one of the addends to make the equation easier to solve. Some students may prefer this strategy as an alternative to left-to-right addition or the breaking up the second number strategy.

Compensation is a useful strategy for making equations easier to solve. More importantly, it encourages students to think flexibly about numbers.

Let’s solve the equation 34+49 using the compensation strategy.

First, since 49 is so close to 50, we will add 34+50. This is easier to solve. Then, since we added one extra to the original equation, we have to subtract one from the final answer.

To see a more detailed explanation of this strategy, and download some free printables to teach it, please see THIS POST.

NEXT STEPS:

Ready to get really strategic with your approach to teaching math facts?

• Implement The Addition Station – a self-paced, student-centered program for the basic addition facts and strategies (see Grades 1-2 HERE and Grades 3-4 HERE).
• Try out Addition Strategies Task Cards as a way for your students to practice each addition strategy in isolation. See the full bundle for 1st Grade, 2nd grade, 3rd grade, or 4th grade.

The post Effective Strategies for Teaching Addition Facts appeared first on Shelley Gray.

WHAT IS A FRIENDLY NUMBER?

In this case, we refer to friendly numbers as a number that is easy to work with. For example, multiples of 10 are “friendly” because they are easy to work with when we add or subtract.

USING FRIENDLY NUMBERS AS AN ADDITION STRATEGY

When we use the “friendly number” strategy for addition, it helps us work with big numbers. This is because we are essentially breaking the equation up into more manageable parts.

We begin by getting to a friendly number, which is typically a multiple of 10, 100, or 100 – depending on the numbers that we are working with. Then we add on the remainder.

Let’s take a look at the “using friendly numbers” addition strategy in action.

EXAMPLES

In this example we will add 27+9 using the friendly number strategy.

First, let’s put the number 27 on our empty number line.

Now let’s get to a friendly number. We know that the number 30 is “friendly” or easy to work with, so we can add 3 to get to 30.

Lastly, we add the remaining 6 and get our answer of 36.

Suppose we are solving 265+18.

First we will write 265 on our empty number line.

Then we can add 5 from the 18 to get to a friendly number 270.

We have 13 left, so now we can simply add the 13 to the 270 to get a final answer of 283.

BIG IDEAS

Mental Math is not about following a one size fits all process or procedure. Mental Math involves being able to think flexibly about numbers and manipulate them in different ways.

For the equations shown above, this friendly number strategy worked well. But students also could have used left to right addition, breaking up the an addend, compensation, or even the plus 7,8,9 strategy.

Our goal is to teach our students to think flexibly about numbers so that mental computation comes easily to them.

NEXT STEPS:

• Download a free activity to practice the using friendly numbers addition strategy HERE.

• Find task cards to reinforce the “using friendly numbers” strategy here:
  • Using Friendly Numbers task cards for fourth grade
  • Using Friendly Numbers task cards for third grade

The post Using Friendly Numbers: An Addition Strategy appeared first on Shelley Gray.

Compensation is a mental math strategy for multi-digit addition that involves adjusting one of the addends to make the equation easier to solve. Some students may prefer this strategy as an alternative to left-to-right addition or the breaking up the second number strategy.

Compensation is a useful strategy for making equations easier to solve. More importantly, it encourages students to think flexibly about numbers.

HOW TO PERFORM THE COMPENSATION STRATEGY

Let’s solve the equation 34+49 using the compensation strategy.

First, since 49 is so close to 50, we will add 34+50. This is easier to solve. Then, since we added one extra to the original equation, we have to subtract one from the final answer.

Let’s suppose that we want to solve the equation 132+64. We can use compensation to add these two numbers. With compensation, there is no one right way to perform the strategy. In this case, let’s begin by taking 4 away from the 64 and add 132+60 to make 192. This is an easy equation to solve. Now, since we subtracted 4 from the original equation, we have to add 4 to the answer. 192+4=196.

NEXT STEPS:

• If you would like full support for teaching addition strategies in your classroom, check out The Addition Station HERE.  These Math Stations are self-paced, student-centered stations for the basic math strategies. Students move through the levels at their own pace, ensuring that they are always challenged, and working to their full potential.
• Read other posts on this website about addition strategies HERE.
• Download a FREE activity for practicing the compensation strategy HERE.

The post Compensation: An Addition Strategy appeared first on Shelley Gray.

Breaking apart an addend is a mental math strategy for addition. Some students may find this method more efficient than left-to-right addition.

This strategy involves breaking up one addend in an equation into more manageable parts. Like many other mental math strategies, this strategy encourages students to think flexibly and to manipulate numbers in different ways. This is the big goal of mental math!

As you look at the examples given, you’ll notice that this strategy reinforces place value understanding, as students are breaking apart the addend by place value.

EXAMPLES

Let’s take a look at how to perform this strategy. Whenever you introduce a new strategy in your classroom, be sure to use small, easy to work with numbers. This will ensure that students can focus on the strategy itself rather than struggling with big numbers while trying to master a new strategy.

In this example, we will add 14+12. We will break the 12 into a 10 and a 2.

Now we add. First we add 14+10 to make 24, and then add the remaining 2 to make 26.

Let’s try another example. Here we will solve 35+46. First we break the 46 into a 40 and a 6.

We will add 35+40 to make 75, and then add the remaining 6 to make 81.

FLEXIBLE THINKING

One of the greatest aspects of mental math is that there is not a series of steps to memorize. Really we just want our students to understand what the numbers mean and be able to manipulate them in a way that works for each individual student.

Suppose we have the equation 213+214.

One student might choose to break up the second number and add 213+200+10+4.

Another student might choose to break up the second number into only two parts and add 200+210+4.

A third student might choose to add these numbers using left to right addition.

Here’s a video with more information on how you can teach the break apart strategy using the Concrete Representational Abstract model:

This is one strategy that you definitely will want to incorporate into your math instruction. It can be used in many different ways and you will notice that your students begin using this sort of thinking for other math concepts as well.

NEXT STEPS:

• If you would like full support for teaching this strategy, find my Break Apart for Addition Unit HERE.

• Download a FREE activity for practicing the breaking up an addend HERE.

The post Breaking Apart An Addend: An Addition Strategy appeared first on Shelley Gray.

Left-to-right addition (also known as front-end addition or the partial sums method) is one of the most powerful mental math strategies for teaching addition of 2 or 3-digit numbers. However, many people are confused by why it is important and why it can be more effective than traditional vertical addition.

WHY IS LEFT TO RIGHT ADDITION SO EFFECTIVE?

The best part about left to right addition is that this strategy promotes real understanding.

When you solve an equation using the standard algorithm (probably the way that you learned to add multi-digit numbers), you use a series of steps. This includes adding the ones first, carrying if needed, then adding the tens, carrying if needed, etc. These steps are committed to your memory, and for those who have excellent memorization skills, this can be effective.

HOWEVER, the standard algorithm does not encourage understanding of place value and number sense. This is the main reason that today’s math instruction tends to shy away from the traditional algorithm in early grades. We want our students to possess REAL understanding of what they are doing. When students are taught methods that encourage mental math, they are able to think more flexibly not only about this isolated concept, but about other math concepts as well.

EXAMPLES

Left to right addition is also effective for adding 3-digit plus 3-digit numbers. In this example, we can see that we add the hundreds first, then the tens, and then the ones. Lastly, we add all of those sums together.

The CRA Model for Mental Math Strategies

NEXT STEPS:

• If you would like full support for teaching addition strategies in your classroom, check out The Addition Station HERE. You’ll find the left to right addition strategy in the Addition Stations for the upper grades. These Math Stations are self-paced, student-centered stations for the basic math strategies. Students move through the levels at their own pace, ensuring that they are always challenged, and working to their full potential.
• Read other posts on this website about addition strategies HERE.
• Download a FREE activity for practicing the left to right addition strategy HERE.

• Check out a Left to Right Addition Strategy Unit” HERE.

•  Find task cards to reinforce the left-to-right addition strategy in isolation here:
  • Left-to-right addition task cards for second grade
  • Left-to-right addition task cards for third grade
  • Left-to-right addition task cards for fourth grade

The post Try Left to Right Addition: A Powerful Mental Math Strategy appeared first on Shelley Gray.

The doubles facts are generally an introductory set of facts that we want our students to become automatic with. We can relate the doubles to so many things around us –  fingers and toes: 5+5, wheels on a car : 2+2, or the eggs in a carton: 6+6.

Our goal for the doubles facts is automaticity. This means that students no longer have to think much about the equation in order to solve it. Rather, they just “know” the answer and are able to say the answer within 1-3 seconds. For example, when a student sees the equation 8+8, he should know that it equals 16 without even stopping to think about it.

Building a strong foundation of doubles will help students with other mental math strategies, particularly the near doubles.

Near doubles involve facts like 4+5. To solve this fact, our students can think, “I know that 4+4=8, and 1 more is 9.”

The goal with the near doubles facts is to build relationships and form connections between facts. We can achieve this by using manipulatives regularly.

Here’s a video that will explain this in more detail:

Teaching the doubles and near doubles facts is important, but it shouldn’t stop simply with the numbers to 12. We need to extend these facts into tens and hundreds as well, and teach our students how we can still use the doubles and near doubles facts in order to solve equations with these bigger numbers.

For example we can use the fact 3+3 to solve 30+30 or the fact 6+6 to solve 60+60=___.

FOCUS ON GROUPS OF 10

To begin teaching students to extend the doubles and near doubles facts, focus on groups of 10. For example, we can think of 40+40 as 4 groups of 10 plus 4 groups of 10. Similarly, to solve 400+400 we can think, “4 groups of 100 plus 4 groups of 100.”

EXTENDING THIS CONCEPT TO OTHER MATH STRATEGIES

This “extending” concept can be used in many other circumstances. For example, when teaching your students the plus 1 facts, teach them to extend that knowledge past simple 1-digit numbers. For example, if we know 8+1, then we can easily figure out 80+10 or 800+100.

NEXT STEPS:

• If you would like full support for teaching addition strategies in your classroom, check out The Addition Station HERE.
• Get full support with the Near Doubles strategy with this complete unit.

• Download a FREE activity sheet for practicing the extending the doubles facts strategy HERE.

• Find task cards to reinforce the extending the doubles strategy here:
  • Doubles and extensions task cards for fourth grade
  • Doubles and extensions task cards for third grade

• Find task cards to reinforce the doubles addition facts here:

The post Extending The Doubles and Near Doubles Facts: An Addition Strategy appeared first on Shelley Gray.

Long division is often considered one of the most challenging topics to teach. Luckily, there are strategies that we can teach to make multi-digit division easier to understand and perform.

The Partial Quotients method is one of these strategies. It is a mental math based approach that will enhance number sense understanding. Students solve the equation by subtracting multiples until they get down to 0, or as close to 0 as possible.

If you would like to try this strategy in your classroom, you may want to start with the Box Method/Area Model, which you can read more about HERE. The Box Method uses the same approach as Partial Quotients, but is organized differently and works well as an introduction.

Let’s learn how to perform the Partial Quotients strategy for long division!

Below, I have included both a video tutorial and step-by-step instructions.

VIDEO TUTORIAL

STEP-BY-STEP INSTRUCTIONS

Suppose that we want to solve the equation 324÷2.

STEP 1:
Write the equation as shown below.

STEP 2:
Now it is time to start figuring out how many groups of 2 will fit into 324. We want to use numbers that are easy to work with, so we try to use friendly numbers like 10, 100, etc. We know that at least 100 groups of 2 fit into 324, so let’s start with that. We write x100 on the right side to symbolize that we are taking out 100 groups. 100 groups of 2 makes 200, so we subtract 200 from 324. This leaves us with 124.

STEP 3:
Now we need to figure out how many more groups of 2 we can make from 124. We know that we can make at least 50 more groups. We write x50 on the right side to symbolize the 50 groups of 2. 50 groups of 2 makes 100, so we take 100 away from 124 and we are left with 24.

STEP 4:
How many more groups of 2 can we make from 24? We know that we can make 12 more groups, so we write x12 on the right side. 12 groups of 2 makes 24, so we take away 24. We are left with 0, which means that we do not have a remainder.

STEP 4:
To find our quotient, we add up the three partial quotients. In this case, we add 100+50+12 to make 162. So 324÷2=162.

Here’s one more example. Let’s solve 679÷5:

We know that there are at least 100 groups of 5 in 679, so we wrote x100. We then subtract 100 groups of 5, or 500, from the 679, and we are left with 179.

We know that there are at least 30 more groups of 5 in 179. 30 groups of 5 makes 150, so we subtract 150 from 179 and are left with 29.

We know that there are at least 5 more groups of 5 in 29. 5 groups of 5 makes 25, so we subtract 25 from 29 and are left with 4.

We cannot make any more groups of 5 out of 4, so 4 is our remainder.

To find our quotient, we add the partial quotients (100+30+5) plus the remainder 4, to make 135 R4.

HELPFUL RESOURCES FOR THE PARTIAL QUOTIENTS STRATEGY FOR DIVISION

I would love to help you teach the partial quotients strategy for long division in your classroom. You may find the following resources helpful:

PARTIAL QUOTIENTS TASK CARDS

These task cards give students the opportunity to practice the partial quotients strategy for long division in a variety of different ways. Students will calculate quotients, solve division problems, figure out missing dividends and divisors, think about how to efficiently solve an equation using the partial quotients strategy, and more. See the Partial Quotients Task Cards HERE or the Big Bundle of Long Division Task Cards HERE.

THE LONG DIVISION STATION

The Long Division Station is a self-paced, student-centered math station for long division. Students gradually learn a variety of strategies for long division, the partial quotients strategy being one of them. One of the greatest advantages to this Math Station is that is allows you to target every student and their unique abilities so that everyone is appropriately challenged. See The Long Division Station HERE.

OR SEE ALL RESOURCES HERE.

The post Partial Quotients: an alternative for traditional long division appeared first on Shelley Gray.

STUDENT #1

This student is knows that 16+4=20 because she has practiced this fact over and over and over again. 

When the teacher asks her to solve 16+5, she’s stumped. This is a fact she hasn’t learned yet.

After moving on to other math units and focusing less on addition, the fact 16+4 is forgotten and will have to be re-learned next year.

STUDENT #2

This next student hasn’t learned the fact 16+4 yet, but can easily figure it out because he knows that 6+4 is ten and then 10 more is 20.

When the teacher asks him to solve 16+5, he is easily able to visualize it because of all the work their class has done with ten frames. He already knows that 16+4=20, so 16+5 is just one more!

Then he thinks of one other way he could have solved it! He imagines the 16 as a 10 and a 6. He adds the 6 to the 5, which is a doubles plus one fact (11) and then adds on the remaining 10 to make 21.

Opponents of mental math will say this is too confusing, but I’ve simply written it all out so that you can see the steps that happen in a person’s head very quickly. This does not need to be written out like this every time.

Near Doubles

This strategy is a natural progression from the doubles. It includes using a known fact and building on it. For example, to solve 6+7, a student could think, “I know that 6+6 makes 12, and one more makes 13.” This strategy is best modelled with ten frames, which makes it easy to see! Students need to SEE math so they can build their understanding!

 

Here’s a video that explains the near doubles strategy in more depth or find a unit for teaching the near doubles facts HERE.

 

 

Make a Ten

Make a Ten is a mental math strategy where students use the number combinations that make ten to form connections and relationships to other facts. First, students must learn the number combinations that make 10. Then they can confidently use those combinations. For example, to solve 8+5, a student might think, “I can take two from the 5 and give it to the 8 to make a ten, and then add the leftover 3 to make 13.” Ten frames are a fantastic way to illustrate this strategy.

 

Here’s a video that explains the make a ten strategy in more depth.

 

Find a unit for teaching make a ten HERE or find the math mats shown below HERE.

 

 

Making Multiples of Ten

This strategy is a natural follow-up to making ten. To make , we can use the number combinations that make ten (6+4, 7+3, etc.). This helps us recognize that expressions such as 26+4 will make a multiple of ten. Ten frames are an excellent way to model the thinking process. For example, when we model 26+4 with a ten frame, it’s easy to see that we can shift the 4 dots over to completely fill three ten frames and make 30.

 

Find a unit for teaching the making multiples of ten strategy HERE.

Left to Right Addition

Left-to-right addition is a powerful mental math strategy for adding numbers with two or more digits. Place value understanding is key, as students will be grouping the tens and then the ones. For example, to solve 24+53, we will first add 20+50 to make 70, then 4+3 to make 7, and finally 70+7 to make 77. Left-to-right addition is important to teach BEFORE students learn the traditional algorithm. This is because left-to-right addition focuses on conceptual understanding rather than on the memorization of a series of steps.

 

Here’s a video that explains left to right addition in more depth.

 

Find a unit for teaching left to right addition HERE.

 

 

 

 

Break Apart/Decomposing

Breaking apart an addend by place value is a powerful mental math strategy for adding numbers with two or more digits. Although this is similar to left-to-right addition, some students prefer it because only one addend is decomposed by place value, rather than both of them. For example, to solve 43+35, we could first decompose the 35 into 30 and 5. We start by adding 43+30 to make 73, then the remaining 5 to make 78.

 

Here’s a video that explains the break apart strategy in more depth.

Find a unit for teaching break apart HERE.

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The post Effective Mental Math Addition Strategies to Master the Addition Facts appeared first on Shelley Gray.

We know that being able to think flexibly is more effective than memorization for math facts. This has been studied in depth by many different math researchers. But sometimes what happens when we begin to teach strategies, is that we “teach” the strategies for memorization. For example, “Whenever you see numbers that differ by 1, you can use the doubles plus one strategy.” Rather than “teaching” the strategies, ideally we want students to discover them and construct their own understanding. The best way to do this is through lots and lots of work with manipulatives. And yes, even if you teach upper grades your students can benefit from manipulatives!

However, sometimes it’s still nice to have a guide for what strategies to steer our students toward.

Seven Different Mental Math Addition Strategies

Below I have outlined seven different different mental math addition strategies that you can model in your classroom to help students build their understanding. I have also included videos and additional resources for some of them.

Counting On – Counting On is a beginning mental math strategy.  Counting on means that you start with the biggest number in an equation, and then count up. For example, in the equation 5+3, you want students to start with the “5” in their heads, and then count up, “6, 7, 8.” This is to discourage students from counting like, “1, 2, 3, 4, 5…..6, 7, 8.” Students also need to understand the commutative property of addition, where if an equation looks like this: “2+6,” they still should start with the bigger number (in this case, 6) and count up “7, 8.”

Here’s a video that will explain this strategy in more depth, or find a unit for teaching the counting on strategy HERE.

Make a Ten – Make a Ten is a mental math strategy where students use the number combinations that make ten to form connections and relationships to other facts. First, students must learn the number combinations that make 10. Then, they can confidently use those combinations. For example, to solve 8+5, a student might think, “I can take two from the 5 and give it to the 8 to make a ten, and then add the leftover 3 to make 13.” Ten frames are a fantastic way to illustrate this strategy.

Here’s a video that explains the make a ten strategy in more depth.

Find a unit for teaching make a ten HERE or find the math mats shown below HERE.

Find a unit for teaching break apart HERE.

I hope this post has helped you make a plan for teaching math strategies in your classroom! I know that if you did not learn this way, it is not an easy transition.

If you’d like more support for teaching addition strategies in the classroom, check out the Mental Math Addition Strategy Bundle that includes units for all of the strategies that were discussed above (photos of contents below).

The post Try These Mental Math Addition Strategies to Boost Math Fact Fluency appeared first on Shelley Gray.