Subtraction (Subtraction Facts)

We also subtract all the time, often without thinking about it.  Someone has 5 pieces of candy and then someone takes two, "Hey, I only have 3 left!"  But, teaching subtraction can be a little trickier than teaching addition.

Definitely try to let students master addition (and learn addition facts) before wading into subtraction.  Not only is subtraction conceptually more difficult than addition, but many students who know their addition facts are able to quickly and easily connect those facts to the subtraction facts in the same fact families.  Not every student makes the connection, but when they do it's amazing.  It definitely worth building the base of addition and trying to help them connect those facts with subtraction facts to cut down on the memorizing that they have to do.

Example:

Student knows that $3+5=8$

The fact family:

$$\eqalign{3+5=8\\5+3=8\\8-3=5\\8-5=3}$$

Even students who know addition facts well will need to learn the concept of subtraction before memorizing fact families, but you can see how, once students know addition facts and learn the concept of subtraction, subtraction facts come together pretty naturally.  See Fact Family lesson for more. 

The easiest way to teach the concept of subtraction is to actually take things away.  This can be done with manipulatives (including fingers) or by drawing dots or lines and then crossing them out as they are subtracted away. One worksheet below (Subtraction with objects) provides drawings that students can literally cross out in order to tangibly subtract.

Example:

$9-5=4$

The student draws 9 circles (or gathers 9 manipulatives or raises 9 fingers).

$$ \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet \bullet $$

Then the student crosses out 5 circles (or removes 5 items or lowers 5 fingers).  

$$ \bcancel{\bullet} \bcancel{\bullet} \bcancel{\bullet} \bcancel{\bullet} \bcancel{\bullet} \bullet \bullet \bullet \bullet $$

The student then counts the remaining items: 4.

This process teaches the concept of subtraction, but unlike "counting on" does not have a logical progression of steps that students can use to become more efficient.  There are a few possible next steps.

Matching with addition facts: as shown above, students who know addition facts and understand subtraction can often map subtraction facts with addition facts and memorize them quickly.  This is the most efficient method but does not work for everyone.

Counting up: Some students develop counting up subtraction strategies on their own, still others find them useful when they are taught them explicitly.  It's not as strange as it sounds.  Essentially, you teach the child to count up from the subtracted number to the whole number -- the space between those numbers is the difference.

Example:

$9-5=4$

The student starts at the subtracted number of 5, then counts up to 9, raising a finger with each count, 6, 7, 8, 9.  There are 4 fingers up, so the difference is four. 

The foundation for this strategy is teaching students that the first number is the number of objects.  The second number removes the first $x$ number of objects: what's left?

So, you have 9, if you take away the first 5, you have, 6, 7, 8, 9,... 4 left.

Counting backwards: Some students count backwards easily.  Other students really struggle to count backwards. By the time students are learning subtraction, they should be able to count backwards.  It's an important part of number sense, of knowing how numbers work.  So, students who struggle to count backwards should work on it.  Students who can count backwards can use if for subtraction.  

For students who can count backwards, subtracting ones and twos is easy.  Subtract 1 is counting backwards by 1.  Subtract two is counting down 2.  But, even problems with larger numbers can use this strategy, although it starts to get cumbersome as the subtracted numbers get bigger (upside, this process works fine with big numbers as long as the big numbers are close to each other).

Example 1:

$9-5=4$

Students hold the number 9 in their heads.  They count backwards, raising a finger each time they count, until they have raised 5 fingers: 8, 7, 6, 5, 4.  They land on 4.

Memorize! In the end, all students should be encouraged to memorize the subtraction facts from 0-20.