So, after releasing my Mental Math addition units on Teachers Pay Teachers, I had many people inquiring about subtraction strategy units. I am SO excited to announce that they are officially FINISHED and posted! I am truly grateful for all of the amazing feedback that I’ve received for the addition units, and I am confident that the subtraction set is just as great!
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.
The subtraction portion of the Mental Math Strategy Collection includes the following units: Counting Back & Counting Up, Thinking Addition, Using Doubles & Building on Doubles, Using Ten, Compensation and Expanding the Subtrahend. Although there are more subtraction strategies out there that you may want to teach, I feel that the ones above 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:
1. 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.
2. 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.
3. 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.”
4. 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.
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.
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.
As I mentioned in my blog post about addition strategies, 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.
One point that I really want to take the time to stress is using mathematical language in your teaching. If you use mathematical language in your teaching, your students will use it too, regardless of their age. In a subtraction equation, the minuend is the largest number, the subtrahend is the number being taken away, and the answer is called the difference. Use these words ALL the time when you look at an equation or write an equation on the board. You will find that your students will begin using these words in no time! Also, teach students to use the term “most effective strategy” or “most efficient strategy” when they are talking about math. When they see an equation, you want them to think about which strategy would be the most effective or efficient. Ask them why. For example, in the equation 8-4, students should realize that using doubles is more effective than counting back.
If you begin to make mathematical discussion part of your everyday routine, you will likely begin to notice students developing their own math strategies, which is exactly what you want to happen! When you have provided them with a solid foundation, students will naturally use what they know to invent other strategies that work well for them. And seriously, how great is that?! Even if you don’t understand a student’s thinking, if it works for them, then that is fantastic! This means that you should not be forcing students to use one specific strategy once they have mastered all of them. Once learning and mastery has occurred, then allow students to use the strategy that is most effective for them.
I hope that this discussion has been valuable for you. If you have any questions about these strategies or need more information on how to teach them, please feel free to comment below or contact me through my Facebook page, “Teaching Resources by Shelley Gray.”
Additionally, here are some links that you may find useful to your mental math instruction:
– Shelley Gray’s Mental Math Strategy Collection on Teachers Pay Teachers (includes addition and subtraction units)
Thanks so much for reading! Have a great day,