Math fact fluency is much more than simply speed and accuracy. True fluency also includes flexibility and appropriate strategy use. We can help students become flexible thinkers who are able to build their own understanding by reinforcing mental math addition strategies.

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.

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.”

Doubles:

The doubles are not necessarily a strategy, but rather a set of facts that is easy to learn and remember. Doubles are all around us; think of fingers and toes – 5+5, wheels on a car – 2+2, or the eggs in a carton – 6+6. Building a strong foundation of doubles will help students with the next strategy, Doubles Plus One.

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 5+6, a student could think, “I know that 5+5 makes 10, and one more makes 11.” This strategy is best modelled with ten frames, which makes it so easy to see! And remember, we have to give our students the opportunity to SEE math so that they can build their understanding!

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.

Making Multiples of Ten:

This strategy is a natural follow-up to making ten. To make multiples of ten, 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.

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.

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.

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.

Subtraction can be a really difficult concept to teach. For me, like many others, it is because I personally am not as comfortable with subtraction as I am with addition. This is most likely true for the majority of your students. Therefore, it is extremely important to equip your students with strategies that will enable them to solve subtraction equations efficiently and effectively. There are many, many mental math strategies that you can teach your students. I have developed units for the ones that I feel are the most important in order to develop a strong foundation for subtraction skills.

I have uploaded the YouTube video below which will help you understand a bit about each strategy, as well as why it is important to teach mental math. Whether you have purchased my units or not, I hope that this video will give you some valuable insight into mental math subtraction strategies. For those of you who are not up for watching a video today, I have included a written description of each strategy below.

Although there are more subtraction strategies out there that you may want to teach, I feel that the ones listed are the most important for developing a strong foundation that will enable students to experience success with subtraction in many different situations. Below is a brief description of each strategy:

Counting Back and Counting Up: 

These are actually two separate strategies, but are so closely related that I decided to include them together in the same unit. Counting back is normally the first strategy that students use when they are learning to subtract. Counting back simply means starting with the minuend (the largest number) and counting back to figure out the difference. For example, in the equation 13-2, a student would think, “13…12, 11” to get an answer of 11. It is very important to remember that counting back is only an effective strategy when the subtrahend (the number being taken away) is a 1, 2, 3 or 4. With subtrahends that are higher than 4, students tend to get mixed up with their counting and get wrong answers. As I mentioned earlier, counting up is closely related to counting back. With counting up, students start with the subtrahend and count up to the minuend. For example, in the equation 10-7, a student would think, “7…8, 9, 10” to get a difference of 3. Counting up is only an effective strategy when the difference between the minuend and subtrahend is a 1, 2, 3 or 4. For example, in the equation 15-5, counting up would not be effective, as students would most likely lose track of how many numbers were being counted.

Thinking Addition: 

Thinking Addition is another strategy that students generally learn early on. Thinking Addition simply means that there is an inverse relationship between addition and subtraction. For example, in the equation 13 – 3, students should think, “What can I add to 3 to make 13?” in order to get a difference of 10. Learning about fact families is an important aspect of this strategy. When students realize that addition equations and subtraction equations can be formed with the same three numbers, they are more likely to use one operation when working with the other.

Using Doubles and Building on Doubles: 

Doubles are some of the easiest facts to remember for many students. When students have achieved mastery with addition doubles, they can use them to solve subtraction equations such as 12-6 or 18-9. Developing automaticity with these facts will cause them to be easily recognizable so that students simply know the fact rather than having to think about it. Building on Doubles means using near doubles in order to solve a subtraction equation. For example, for the equation 15-7, students should think, “I know that 15 is 1 more than the double of 7, so the answer must be 1 more than 7.”

Using Ten: 

Ten is otherwise known as a “friendly number” in mathematical terms. This is because it is an easy number to work with. With the ‘using ten’ strategy, students learn to substitute ten for another number, and then later adjust the answer. For example, in the equation 13-8, students will first think, “10-8” to make a difference of 2, and then add on the remaining 3 that was taken from the original equation to make a total difference of 5.

Compensation: 

Compensation is one of my favorite subtraction strategies, because once students know how to compensate, they can begin using it in many different math situations. Compensation is most effectively used when the subtrahend of an equation is either an 8 or a 9. When using compensation, students change the subtrahend to a 10, and then adjust the answer. For example, for the equation 15-9, students will first calculate the difference to 15-10 to make 5. Then, because 1 too many was subtracted from the original equation, 1 must be added to the difference to make 6.

Expanding the Subtrahend: 

Expanding the subtrahend is a valuable strategy when working with multiple digit equations. Students use expanded notation for the subtrahend, and subtract it in two steps. For example, in the equation 45-21, students first perform 45-20 to make 25, and then subtract 1 more.

It is important to think of each strategy as a tool that you are giving each student for his tool kit. Once students have all the tools, they can decide for themselves which one will do the best job. However, in order to choose the most effective tool, they must have a really good understanding of how each one works. This is why it is so important to help students develop mastery at one level before moving onto the next.

Mastery means that students can perform the strategy efficiently, without confusion and uncertainty. To develop mastery, it is important to teach each strategy in isolation before integrating them together and expecting students to choose the best one for the job. This means that if you are beginning with the counting back strategy, you should teach it in full until students are completely comfortable with it. Then, once they are comfortable and you feel that they have achieved mastery, it is time to move on to counting up, while still reviewing counting back. Once mastery has been achieved with respect to counting up, students are ready to learn about thinking addition, and the process continues. Your ultimate goal when teaching mental math strategies should be to get students to the point where they can look at an equation and use an effective strategy to solve it. However, they will not be able to do this unless you allow them to achieve mastery of each strategy and use it extensively before moving onto another one.