# Teaching three-year-old subtraction using the number line

I am aware of questions such as this one. On the other hand, I still believe that teaching a bright three year old subtraction is possible. He counts from $$0$$ to $$100$$ and backwards from $$10$$ to $$-10$$, or some more negative integer. What I did was using a number line such as the following:

Say I want to teach $$3-5$$, I will ask the child to point his finger at $$3$$, then move $$5$$ units to the left, then ask the child where his finger is now. He will reply "negative two" and I will write down the equation $$3-5=-2$$. Here the minuend is not necessarily larger than or equal to the subtrahend.

My question is, is the a right way to teach a three year old subtraction? I personally find that early exposure to negative numbers exciting but am not sure if it could be counter-productive or "haste makes waste".

It seems to me that most people teach by asking questions such as "Here are five apples, if I take away three, how many are there left"? However, I find this method limited as the child may have the misconception that in $$a-b$$, $$b$$ cannot be larger than $$a$$. Also, it seems to me that the usage of the number line reflects the spirit of more advanced mathematics.

Edit: When I asked this question about 10 months ago when the student is only a little over 3, she does not seem to get the number line idea well. Now as she is about to turn 4 (though technically she is still 3), she understands it and enjoys doing calculations such as $$5-9$$ using the number line.

• My brother taught me negative numbers when I was 5. It made me feel special to know about them and I went on to major in math. If the child is interested that's fine. However, at some point he will learn about taking 3 apples from 5 apples, and the teacher will tell him, you can't take a bigger number from a smaller number Make sure you prepare him for this misinformation. – Amy B Feb 18 '19 at 13:22
• @AmyB, "the teacher will tell him, you can't take a bigger number from a smaller number". I wish they say "you can take a bigger number from a smaller number and you will learn how when you are older" – Zuriel Jan 3 '20 at 20:41

## 3 Answers

The following is anecdotal and rambling, but I think it goes a good way towards addressing the question (from a personal point of view).

My daughter and I play a variant of the card game War to work on arithmetic facts. Originally, we played "Addition War": we would each play a card, and the first to sum the two numbers got to keep both cards. When her first grade class started drilling subtraction facts, we started playing "Subtraction War." At one point, she turned up a 5 and I turned up a 3. She got very angry with me when I said "Oh, three minus five is negative two!"

"Papa!" said, "You can't minus five from three! Five is too big!"

At this point, I introduced her to a number line model of addition and subtraction. The story that I gave her is that a frog is sitting on a lily pad in a pond. Every lily pad has an address or number, counting up from 1, going from left to right. When we add, the first number tells us where the frog starts, and the second number tells us how the frog hops. For example, $$3+4$$ means that the frog starts at the lily pad labeled "3", then hops to the right four times, which takes it to the lily pad labeled "7". Therefore $$3+4=7$$.

(Note that this was all done on a piece of paper with an actual toy frog to hop around, which makes life a little easier.)

Once she got the idea of addition with the frog, we started playing the same game with subtraction. The only difference is that the frog now moves left instead of right. For starters, we only worked "proper" subtraction problems, where the minuend is larger than the subtrahend. I then asked about $$4-4$$. Our lily pads started at 1, so this didn't seem to make sense (the frog splashed into the water). However, since we are clever mathematicians, we can plant a lily seed and grow a new lily pad. She had already seen zero before, so it made sense to call this new pad zero, and get on with life.

Finally, we can introduce negative numbers. I got out a different color, and started "planting" more lily pads to the left of zero. I labeled them going up from right to left in the new color (which happened to be purple). She was really resistant to this kind of labeling at first—she didn't like there being two different "3s", for example, but we managed to talk about "left three" and "right three", and she eventually bought the idea. After a bit more play, she got pretty comfortable with the idea of problems like "Five minus nine is 'purple four'" (or 'left four'). A few months later, she is pretty comfortable with the idea of (small) negative numbers.

Now when we play subtraction war, black cards always come first, so the "correct" difference if I play $$5\spadesuit$$ and she plays $$6\heartsuit$$ is "Negative one!" If the cards have the same color, either positive or negative answers are fine.

In any event, I think that this model of subtraction—which really is exactly what you propose—is not only appropriate for an introduction to subtraction, but is, perhaps, the "right" model for addition and subtraction in general. It might seem less concrete than "three apples plus two more apples is five apples," but I think that it gives a much more concrete meaning to negative numbers. Frankly, I still have trouble with negative quantities (negative apples scare me), but I have no problem with negative coordinates or negative directions.

You might need to use both, the number line, and the apples. It is always interesting to have a visual idea of a concept, because raw numbers sometimes aren't really intuitive. To get away from the limit where you can't take more apples than the number of apple you have, there is a simple trick.

You have to give me 5 apples, but you only have 3 apples. So, give me the first 3 apples, you have now zero apple. I need 2 more apples, but i'm a nice guy, so you can give me those two apples later. You now have negative 2 apples, which means, if you find apples, the first two apples you find belong to me, not to you.

• Interesting! I am not sure if "negative 2 apples" is too complicated for my student but she seems doing fine with the number line. – Zuriel Jan 3 '20 at 20:40

Zuriel, START with the apples version. Yes, it is not as good as the number line. But it gets you STARTED.

You are dealing with a 3 year old. Build muscles gradually. Would you teach a forward roll first or a front flip?

There is this urge, urge, urge on this forum to want to jump to "cleaner" but more advanced formulations. This shows a total misunderstanding of human psychology.

• How has it worked in your experience? – Tommi Feb 18 '19 at 7:38
• Is this based on research, your own experience, or is just a rant against pushing kids before they're ready? Do you have experience with gifted children? In my 25 years of working with them, I have found that they appreciate jumping ahead. – Amy B Feb 18 '19 at 13:24