# PUFM 1.4 Subtraction

Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”

Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.

## The Advantage of Number Bonds

This connection between addition and subtraction is represented in many textbooks by the “four-fact family.” The idea is that if students know one of the facts in the family, then they know all of them. Many students never see the connection, however, and think of these equations as four separate little bits of abstract information, all of which have to be memorized. This can overload their minds and make them give up on math.

A four-fact family looks like this:

4 + 2 = 6
2 + 4 = 6
6 – 4 = 2
6 – 2 = 4

Number bonds connect to the student’s understanding at a deeper level, showing all four of the fact family relationships in a single picture. You can use either the circles (like a pile of small items which can be pulled apart and then slid back together) or a bar model diagram — whichever you prefer.

## Study Teaching Materials

In this section, our textbook only refers to Singapore Primary Math 3A pp. 18-23. It seems to me that the following would also be helpful, if you have these books: 1A pp. 38-51 and 65-67; 1B pp. 10-13, 32-39, 82-87, and 94-96; 2A pp. 22-51; 2B pp. 6-19; 3A pp. 18-23 (which introduces the Singapore math models or bar model diagrams), and 36-38. Page numbers are from my 3rd edition books. If you have a different edition or another textbook series, look for headings like these:

• Subtraction
• Comparing Numbers
• Mental Calculation
• Word Problems

## Many Ways to Understand Subtraction

With two models (set and measurement) and three interpretations of subtraction, we have a total of six situations to represent in story problems. Remember that the set model refers to things that are discrete and countable, such as strawberries and stuffed animals. The measurement model refers to things that are continuous and easily split into smaller pieces (including fractional amounts), such as fabric or fruit punch.

As with other topics, our students need to think their way through lots and lots and lots of word problems of all sorts, so they learn to recognize subtraction in all its disguises.

Can you come up with a word problem for each type of situation?

• Part-Whole Interpretation:
(Set model) Part of group A is X. How many are not-X?
(Measurement model) Part of measurement B is Y. How large is not-Y?
• Take-Away Interpretation:
(Set model) Some X of group A is taken away or used. How many are left?
(Measurement model) Part Y of measurement B is taken away or used up. How much is left?
• Comparison Interpretation:
(Set model) How many more in set A than in set X?
(Measurement model) How much larger is measurement B than measurement Y?

Notice that the part-whole interpretation is the most general. Both the take-away model and the comparison model can be seen as special cases of part-whole:

• One part is taken away, and one part is left.
• One part is the smaller amount, and one part is the difference.

The comparison interpretation is by far the most difficult for children. The other situations give them some sort of story to guide their imaginations, but in a comparison, nothing happens — nothing is taken away or changed, so there’s no action to guide a child’s thinking. If your children get stuck, try modifying the question. Instead of asking who has more, ask something like, “How many would we have to give Tommy so he would have the same number as Annika?”

## Bar Model Diagrams

With subtraction, we see the first significant advantage of Singapore math models, which are commonly called “bar diagrams.” Bar models make it easy to see comparison relationships, which are more abstract than “take away” situations and therefore more difficult for children to wrap their minds around.

Bar model diagrams will continue to be powerful problem-solving tools as our children work their way through the elementary curriculum. If you have students who thrive on hands-on work, you might consider using Cuisenaire rods as a manipulative to help students develop an intuitive feel for bar diagrams.

Here’s a fun Cuisenaire rod game for practicing number bonds:

Eight is Having a Party!

## Thinking Strategies for Subtraction

Beginners solve simple subtraction problems by counting backward, but we want to help our children progress to more conceptual thinking strategies. All of these mental math strategies look more difficult when written out than they are in real life. When children practice them over and over, they get very fast. Sometimes it’s hard for my old brain to keep up.

Remember what I said in the last post: Our goal at this level is NOT for our children to memorize a series of math facts, but to develop confidence in working with numbers. If we stress fact memorization too early, we will short-circuit the child’s learning process. Once children “know” an answer, they don’t bother to think about it — but it is in the “thinking about it” stage that they build a logical foundation for understanding all numbers.

Let’s examine several different ways to think about the calculation:

$54 - 18 = \, ?$

Counting Down
The goal of this strategy is not simply to count backwards, but to take away the number in easy chunks. First take away the easy 10 (leaving 8 more to take away). Then how many do we take away to make 40? And how many are left to take away after that?

• $54 - 18 = (54 - 10) - 8$
$\rightarrow 44 - 8 = (44 - 4) - 4$
$\rightarrow 40 - 4 = 36$

Counting Up
This strategy uses the definition of subtraction as a “mirror image” of addition. How many would we add to 18, to make 54? First add 2 to make 20, then add 34 more, so we’re adding a total of 36.

• $54 - 18 = \, ?$
$\rightarrow 54 = 18 \, + \, ?$
$\rightarrow 18 + 2 + 34 = 18 + 36$

Make It Easier (Compensation)
“Simplify the amount subtracted.” When we add the exact same amount to each number, the difference between the numbers stays the same, but we can make the actual calculation a whole lot easier. Since 18 + 2 = 20, and that will be easy to subtract, we can add 2 to BOTH of the numbers without changing the difference between them.

• $54 - 18 = (54 + 2) - (18 + 2)$
$\rightarrow 56 - 20 = 36$

If Only, If Only…
This strategy is another version of compensation, and I like it better than the one above. We find an easy number to take away, then adjust to make our answer correct: “If only we were taking away 20, that would be so much easier!” But we took away too much — what should we add to balance it?

• $54 - 18 = 54 - 20 + 2$
$\rightarrow 34 + 2 = 36$

## Place Value Considerations

Remember, place value is “invisible” to us as adults, so we have to pay special attention to these issues. For mental math, when we work the bigger parts (tens or hundreds) first and the smaller parts (ones) last, our answer is much easier to keep in short-term memory because it appears in the order we normally say it.

## Homework Set 4

My favorite homework problem in this set is #6, making up word problems. In fact, I recommend going through any homework page that is merely calculations and making up stories to represent each problem. Remember to use the different models (set/measurement) and interpretations (part-whole/take-away/comparison) in your stories.

• Better yet, take turns with your child to make up stories, and then write down your favorites in your math journal!

My least favorite homework question was #7c. Why would anyone ever think of different strategies for this problem? 8 + 10 = 18 is one of those nearly-automatic calculations based on place value, and there is no reason in the universe to think about alternate strategies for it. Doing so can only confuse our students.

I think the question is a typo or editing error. They probably meant to write something like: “List as many different strategies as possible that can be used to calculate $103 - 78$ .”

• How many strategies can you think of?
• Which of them is your favorite?

This post is part of the Homeschooling with a Profound Understanding of Fundamental Mathematics Series. [Go to the previous post. Go to the next post. Or start at the beginning.]