Now we arrive at the concept of partial numbers. When I teach a child about partial numbers, I find that the most helpful image is pies — banana cream, cherry, whatever, it doesn't matter what kind of pie. Why pies? Because they are easy to draw (it's just a circle) and easy to cut (it's just a line through the circle). Two and a half pies is just two circles and a half circle and children grasp the concept easily, especially when you ask, "Where did the other half pie go?" and answer, "You ate it, didn't you?" Pies taste good and are pleasant thoughts, just the sort of thing we need to keep a child's attention focused on the task at hand: learning partial numbers.

Forgive my repetition, but learning is not only understanding what a thing is but also what it isn't, and when teaching partial numbers, applying that concept pays off in spades. It will often happen, for instance, that when you think you have done a good job teaching your child fractions and she insists that she understands them, you will ask her to multiply two fractions and she will start by finding their common denominator, which of course is wrong.

What's going on here? Here's what's happening. It's not hard to learn a rule, what's hard is putting that rule somewhere in your mind so that you can use it when you're supposed to and only when you're supposed to. The problem is that there are so many rules that they get confuddled. It's not hard, for instance, to know how to find a common denominator, what's hard is knowing when to find a common denominator and when not to. For that, we need order, an arrangement of some sort, like a wall of hanging hooks so that each new idea has a proper place to hang. What's needed is a context, a background, so that each idea has a proper place so that it can all be seen together and the shape (topology) of the whole looks right in the mind's eye.

How do we do that? The best way to keep track of the rules is to show her the big picture with a picture, so she can see all of it up front, so she’ll know what all really is. And here's how I do it.

The 16 Things

There are 16 things to learn in arithmetic. (Actually there are more but these 16 are the lion's share.) The 16 things are:

Add Whole Nums	Subtract Whole Nums	Multiply Whole Nums	Divide Whole Nums
Add Fractions	Subtract Fractions	Multiply Fractions	Divide Fractions
Add Mixed Nums	Subtract Mixed Nums	Multiply Mixed Nums	Divide Mixed Nums
Add Decimals	Subtract Decimals	Multiply Decimals	Divide Decimals

Now that doesn't look so daunting, does it? Or maybe it does. But armed with this checklist, it is much easier for a child to remember which rules apply to which thing. It’s not just what to do but when to do it.

But she needs more than a checklist. She needs a checklist where she can write the rules herself, and sample problems, because by writing, children remember. We remember what we write. Here is what my expanded checklist chart looks like. Draw with a pen (or use Excel if you must) this 5 by 5 table on a full sheet of paper so that each cell has plenty of room to write it:

	Whole Numbers	Fractions	Mixed Numbers	Decimals
+
−
×
÷

I take the whole sheet to draw this table with lots of room in the cells to write. The point of the table is not to document operations but to give her a place to write what she is leaning so that she can see what goes where and how it all knits together. A picture is worth a thousand words, but if she draws the picture, it's wroth a million words. And that's just what we're going to do. You will print or draw (and eventually she will draw) this table over and over again. With this table she will review constantly so that at the end, processes don't get mixed up with other processes — for instance, she will remember that common denominators are for adding and subtracting fractions and not for multiplying and dividing fractions. Okay, let's begin.

Whole Numbers: Adding, Subtracting, Multiplying, Dividing

We've already covered all of this, that’s how we got to where we are now. But now she will understand that arithmetic with whole number is only half the battle. Whole numbers (one pie, two pies, ...) are one kind of thing and part numbers (a half a pie, a third of a pie, two thirds of a pie, ...) are another kind of thing. And that is where we are leading her brain now.

To acquaint her with the table, it would be a good idea to review whole numbers by writing four problems ( + − × ÷ ) in the four cells of the Whole Numbers column, and of course have her solve them.

Fractions: Addition

Here is where our pies come into play in earnest. As I said, a picture is worth a thousand words, but if she draws it, it's worth a million. So, when you need pies drawn, when it makes sense, give the pencil to her.

Now, to add and subtract fractions she needs to first find a common denominator. The mechanics for doing that are not too difficult but understanding why we need a common denominator is challenging. Here's how I explain it —

Me: If you add 2 apples to 3 oranges what you have?
Her: I don't know.
Me: You have 2 apples and 3 oranges.
Her: Pardon? What's the point?
Me: Let's try this instead. If you add 2 apples to 3 apples what do you have?
Her: 5 apples.
Me: Good. So you can add apples to apples but you can't apples to oranges. Do you see why that's true?
Her: I understand. You can't add things that are not the same.
Me: That's right. And fractions are like that. You can't add one half to one third (I write ¹⁄₂ + ¹⁄₃) any more than you can add one apple to one orange. One half is like one apple and one third is like one orange. We could add an apple and an orange if we could magically change them both to bananas but we can't do that. But we can change the half and the third to the same number so that we can add them. And that same number we call a common dominator. And that is what you are going to do right now. Are you ready?

Now that she understands why we need a common denominator, let's teach her how we get a common denominator and what we do with it. Let's stick with our simple example of one half and one third. How do we find a common denominator? There are two ways.

The first way is to multiply the two denominators. In this case, 2×3=6 so 6 is a common denominator.

The second way is to write Count By lists. For one half we have 2,4,6,8,10,12,... and for one third we have 3,6,9,12,... What number appears on both lists? There are two. 6 and 12 are on both lists so we have our pick. Which should we pick? We should pick the smallest because that makes things easier. So it's not just a common denominator we want but the smallest one. We call that the Least Common Denominator or LCD.

So now that we have our Least Common Denominator, 6, what do we do with it? We rewrite our fractions, like so.

¹⁄₂ + ¹⁄₃ → ^?⁄₆ + ^?⁄₆

Our two new fractions both have a denominator of 6, thus they are "common" (the same). But we don't know what the numerators are yet, right now they are just questions marks and we have to figure out what they really are. Until we find out what those numerators are, these new fractions are, in a sense, broken, and we have to fix them by discovering what those new numerators are.

How do we find those numerators and fix the fractions? By doing the same thing to each numerator that we did to its denominator. So, what did we do to the denominators? How did we change the 2 into a 6, and how did we change 3 to a 6? We multiplied the 2 by 3 to get 6, so that is what we have to do to the numerator as well. And we multiplied the 3 by a 2 to get 6, so that is what we have to do to that numerator. And it looks like this —

1 × 3 = 3 so ¹⁄₂ → ³⁄₆

and

1 × 2 = 2 so ¹⁄₃ → ²⁄₆

so

¹⁄₂ + ¹⁄₃ → ³⁄₆ + ²⁄₆

Our problem now has a common denominator, 6, which we can now solve easily.

³⁄₆ + ²⁄₆ = ⁵⁄₆     just like 3 bananas + 2 bananas = 5 bananas.

And that's how to add fractions.

Fractions: Subtraction

Now, what about subtraction? Our first two steps are exactly the same as with addition: we find the common denominator and fix the numerators. But then, we subtract instead of add. Let's continue using the same example: ¹⁄₂ and ¹⁄₃.

¹⁄₂ − ¹⁄₃ → ³⁄₆ − ²⁄₆ = ¹⁄ ₆

because 3 − 2 = 1.

But there may be one additional step. Let's not forget the Subtraction Flip Rule. If the number we are subtracting is larger than the number we are subtracting from, we must flip them, do the subtraction, then place a negative sign (−) in front of the answer. This is true of all subtraction — whole numbers, fractions, mixed numbers, and decimals. So let’s change our example to something more challenging. Suppose the problem is —

¹⁄₃ − ¹⁄₂

This is harder because we get —

¹⁄₃ − ¹⁄₂ → ²⁄₆ − ³⁄₆

Now we want to subtract. But wait a minute. We can’t subtract, can we? Why? Because we can’t subtract 3 from 2.

Well, actually we can. We just have to be clever, and remember the Subtraction Flip Rule. So that, rather than

²⁄₆ − ³⁄₆ = ?

we instead flip it to get —

³⁄₆ − ²⁄₆ = ¹⁄₆

which is the right number but is not quite right yet. Because we flipped it, we need to now unflip it and put a negative sign in front of the answer like so —

²⁄₆ − ³⁄₆ = − ¹⁄₆

which is the correct answer. And that is all there is to subtracting fractions, except that now your student needs to do this again and again with more and bigger examples.

Fractions: Multiplication

Multiplying fractions is easier than adding fractions. You do not mess with a common denominator, in fact, that hardest part is remembering that you don’t mess with common denominators. All you do is multiply the two numerators and multiply the two denominators. For example —

¹⁄₂ × ¹⁄₃ = ¹⁄₆

For another example —

²⁄₅ × ⁵⁄₇ = ¹⁰⁄₃₅

Now this example requires one additional step. Since 10 and 35 are both divisible by 5, we should do that to simply the fraction. We call that "reducing." So —

²⁄₅ × ⁵⁄₇ = ¹⁰⁄₃₅ (÷ ⁵⁄₅) = ²⁄₇

And that's all there is to it.

A general rule: Whenever we can reduce a fraction, we ought to. In that last example, ¹⁰⁄₃₅ is a correct answer but, in a sense, not correct enough. It's not that ²⁄₇ is more correct, it's that it is better because it is simplier. And on a test, a teacher may mark an unreduced fraction as wrong.

Fractions: Division

Dividing fractions is almost as easy as multiplying fractions, but there is one additional step, a flip rule. Not the Subtractions Flip Rule but the Division Flip Rule. To divide fractions you flip the divisor (or more properly, you reciprocate the divisor) then multiply the fractions. For example —

¹⁄₂ ÷ ¹⁄₃ → ¹⁄₂ × ³⁄₁ = ³⁄₂

And so we have a correct answer, ³⁄₂. However, ³⁄₂ is not the best answer, and that's because it is a improper fraction; that is, the numerator is larger than the denominator. And so, again in the spirit of simplifying, we want to convert this improper fraction to a mixed number. And the way we do that is to divide 3 by 2 which gives us —

¹⁄₂ ÷ ¹⁄₃ → ¹⁄₂ × ³⁄₁ = ³⁄₂ = 1¹⁄₂

And that is how it's done.

The general rule expanded: Always we want to simplifly, to make things easy whenever possible. And now we have two rules to simplify fractions: First: Reduce if you can; that is, if the numerator and denominator are divisible by the same number (meaning integer), than do it. And Second: Convert improper fractions to mixed number; that is, if the numerator is larger than the denominator, divide the numerator by the denominator, and that is your final answer.

Now, return to the checklist chart and have her write in the steps in the fraction column. For instance, for addition she might write “common denominator” or just C-D for a member lock, and “add numerators,” and maybe provide an example. For subtraction, she might write, "same as addition" except she must also remember the Subtraction Flip Rule. For multiplication, she must remember, “NO common denominator.” And for division she must also say "NO common denominator" and remember the division flip rule. But my point is that her writing it onto the chart will cause her to remember.

She may want to keep the chart. My opinion is that is a bad dependence. It goes in the trash, unless she is adament to keeping it, in which case you give in, of course. But I get my students to crinkle and toss it thus breaking their dependence on a piece of paper. She’s going to write it all again tomorrow anyway and that’s how she’ll remember, not by keeping a piece of paper that she’ll constantly have to refer to.

Oh, and it doesn't need to be neat — you're going the throw it away anyway and write it again tomorrow — but neatness does matter. It doesn't do much good to write if you can't read what you write. So writing on the chart not only practices math but it practices neatness as well.

Mixed Numbers: Add, Subtract, Multiply, Divide

A mixed number is a whole number plus a fraction. For example, 2³⁄₄ means 2 plus ³⁄₄, that's why we say, "2 and ³⁄₄", and means plus. To draw it for a child, it’s 2 pies (2 circles) and ³⁄₄ of a pie, which you’d draw as a circle with a gaping mouth where the ¹⁄₄ piece is missing, like what Pac-Man used to look like.

To do arithmetic with mixed numbers, she must first convert the whole number part to a fraction then add the two fractions so that we end up with only fractions and no whole numbers. Then, once was have only fractions, we can add, subtract, multiply, and divide them just as I’ve explained above. Let's use this example:

1¹⁄₂ + 2²⁄₃

We first convert the whole numbers 1 and 2 to fractions. But not just any fractions, but fractions with the same denominator as their ajoining fractions (so we have common denominators so that they can be added. Like so:

1¹⁄₂ becomes ²⁄₂ + ¹⁄₂ which becomes ³⁄ ₂

And also,

2²⁄₃ becomes ⁶⁄₃ + ²⁄₃ which becomes ⁸⁄ ₃

And now our original problem has become

1¹⁄₂ + 2²⁄₃ → ³⁄₂ + ⁸⁄₃

So because we have converted our two mixed numbers to fractions, we are now adding two fractions. And what is the first step in adding two fractions? Finding a common denominator, which in this case is 6 (2 × 3 = 6) and our problem becomes

1¹⁄₂ + 2²⁄₃ → ³⁄₂ + ⁸⁄₃ → ⁹⁄₆ + ¹⁶⁄₆ → ²⁵⁄₆

Finally, we convert our improper fraction to a mixed number by dividing 6 into 25 to get

²⁵⁄₆ = 3¹⁄₆

And that is how you add mixed numbers.

I will point out that once you’ve converted the original mixed numbers to pure fractions, you're not limited to just adding them, you can also subtract, multiply, or divide them according to the rules of fraction arithmetic I given you.

Now, to be thorough, there is another way to add and subtraction mixed numbers, and that is to add or subtract the whole numbers first and the fractions separately. And yes, that works, for addition and subtraction, but not for multiplication and division. So, if your munchkin learns how to do only what I’ve just showed you, that one method works for all four operations, and since it’s one method only, it’s easy to remember. And if she can do that, she’s learned a lot! As far as that other method of adding and subtracting mixed numbers are concerned, yes, she will have to learn to do that because schools will insist on it. That's not a bad thing, but as far as practical math is concerned, just getting it done and finding the right answer, what I've just showed you gets the job done.

Now, back to the chart. It's time for her to remember what she's just learned. Have her write down some rules for adding, subtracting, multiplying, and dividing mixed numbers, and some examples. And, of course, throw it away — unless she really insists on keeping it, but only ‘till your next session.

Decimals: Introduction

Decimals is another way of writing partial numbers. Every kid gets befuddled by decimals and wonders why bother. When I was in the sixth grade (a looong time ago) my buddy Tommy (an A student) ridiculed decimals with "dis smells". But we have to learn decimals if for no other reason than its how we handle money.

The reason we use decimals is because decimals are more economical (faster, breezier, less writing) than fractions. As long as we restrict ourselves to denominators of 1, 10, 100, 1000 and so forth, decimals make life easier. With decimals, we do not write the denominators because there is no need to write them because we already know what the denominators are by the number of places to the right of the decimal point. For instance, .3 is ³⁄₁₀ and .03 is ³⁄₁₀₀.

The problem with decimals is that the economy comes at a cost which is that while every decimal can be expressed as a fraction, not every fraction can be expressed as a decimal. For instance. ¹⁄₃ cannot be expressed as a decimal value, at least, not absolutely. We can try, we can write .333333... but we will soon give up when we realize that we can't write an infinite number of 3s. And we can cheat by using an underline symbol, as .3, to mean "imagine, if you can, that there is an infinite number of 3s after that decimal point."

The problem is that .3 (³⁄₁₀) is not ¹⁄₃ although it's close. And .33(³³⁄₁₀₀) is not ¹⁄₃ although it's closer. And so forth. So we just accept that for the convenience of decimals (and decimals are very convenient) we can write some numbers and not others. But that's okay, especially when we're talking about money.

One implication of all this is, suppose you had to solve a problem that used both fractions and decimals. For instance if you saw ¹⁄₃ + .5 — how would you proceed? It's highly unlikely you'd ever see such a problem in the real world, but on an IQ test at a job interview, you might. So, what to do? Clearly you must either convert the fraction to a decimal or the decimal to a fraction. So, which is better? It's better to convert the decimal to a fraction so you can avoid getting entangled with .33333333333... or something of that sort. With fractions, you’re always on solid footing. With decimals, you’re never sure.

Now, let us begin.

Decimals: Addition

When you add or subtract fractions, recall that you must find and use a common denominator. When you add or subtract decimals you do the same thing but in a different way: you line up the decimal points. For instance, to do this problem

12.34 + .5678

we must make it look like this:

12.34
+  .5678

To be thorough, we can tidy it up by adding zeros to the right of the top number without changing its value, so we get:

12.3400
+  .5678

By lining up the decimals we have found our common denominator which in this problem is 10,000 and the problem really is

12³⁴⁰⁰⁄₁₀₀₀₀ + ⁵⁶⁷⁸⁄₁₀₀₀₀

But that's not important to know. All you need to know is: line up the decimal points, remember to bring the decimal point down into the answer, and adding decimals is easy. It's certainly easier than finding common denominators, that's part of the economy of decimals. Now we do the problem just like any other addition problem. All that’s different is that we lined up decimal points instead of finding common denominators. But actually, they’re the same thing. In any case, our problem finishes up like this —

12.3400
+  .5678
12.9078

And that's all there is for adding decimals.

Decimals: Subtraction

Subtracting decimal number is no different than adding: you must line up the decimal points then subtract. But there is one more thing to remember, and that is, of course, the Subtraction Flip Rule: If the bottom number is bigger than the top number, flip them, subtract, and sick a negative sign (−) in front of the answer.

Decimals: Multiplication

The first thing to remember with multiplying decimals is don't line up the decimal points, just leave them where they are. (This is the same as the first rule with multiplying fractions: don't find a common denominator.) Just do the multiplication and pretend for the moment that the decimal points are not even there. For example:

     12.3
   ×4.56
      738
    6150
+49200
  56088

But now, where does the decimal point go? You add up the decimal positions; that is, the total number of places to the right of the decimal places in the problem. In 12.3 there is one decimal place (.3). In 4.56 there are two decimal places (.56). So altogether there are three (one+two) decimal places and that is where you place the decimal point in the answer so that 56.088 is the final answer. So, take your pen (or pencil) and put it there —

     12.3
   ×4.56
      738
    6150
+49200
 56.088
     ^

And that's all there is to multiplying decimals.

Decimals: Division

A quick review first. A division has three numbers: a dividend (the number to be divided), a divisor (the number that the dividend is divided by), and the quotient (the answer). There are four ways we can express a division problem.

First: we can say: "ninety-six divided by three equals thirty-two."
Second: we can use the divide operation symbol: 96 ÷ 3 = 32
Third: we can use a fraction: ⁹⁶⁄₃ = 32
Fourth: we can use a divide box:

   32
3|96

All four of these mean exactly the same thing and are used interchangeably. But of the four, only the last one, the divide box, gives us a method (a paradigm) for actually figuring out the answer to the question. In this example the question is: well, what is 96 divided by 3? How we proceed to get the answer (32) is the subject of another article, the one I wrote on Division, so I needn’t repeat it here.

Then why the review? Because to understand how to divide decimals, you must understand how decimal points float; that is, how you can move them to the right or left without "breaking" the problem.

Here's the rule: In a division problem, you may move the decimal point of the divisor to the right or left as many places as you want if you also move the decimal point of the dividend the exact same number of places in the same direction. Here are some examples.

12.34 ÷ 5.6 is the same as 123.4 ÷ 56 which is the same as .1234 ÷ .056

and

^12.34⁄_5.6 is the same as ^123.4⁄₅₆

and most importantly for our subject here

     _____                               ______
5.6|12.34  is the same as  56|123.4

And that brings us to this question: How do we solve a problem like this?

      _____
.12|.36

First, you get rid of the decimal point in the .12 then you can do the division. But how do you safely get rid of the decimal point? You move it two places to the right just after the 2 so that .12 becomes just 12 which is now a whole number. But to do that safely you must do the very same thing to the .36 and also move its decimal point to the right so that .36 becomes just 36 which is also a whole number. So we see that

      __                             __
.12|.36 is the same as 12|36  and now we can do the division to get 3.

But that's too simple. Let's do a more difficult and more typical problem, where there is a decimal point still remaining in the dividend. There is an important rule which is simple: Wherever the decimal is in the dividend, it must also be in the quotient in the same place just over the decimal place in the dividend. In fact, I have my students write that decimal point in the quotient before they work the problem. So, let's do this problem:

     ______
.12|3.648

First we move the decimal point two places to the right (in both the divisor and the dividend) to get:

    ______
12|364.8

So we have removed the decimal from the divisor, but it is still in the dividend, just in a new location. Since it is in the dividend, the decimal must also be in the quotient in exactly the same position.

Now we can do the division like any other and we get:

       30.4 
12|364.8
   -36
       04
        -0
         48
        -48
           0

And that is how you do long division with decimals.

Except, what about this problem?

     _____
.12|364.8

Oh, well, that's different. We want to move the decimal two places to the right, but the dividend doesn't have two decimal places, it has only one. So, what do we do? Easy. Just stick a 0 on the end of the dividend so that it does have two places, then move your decimal point. Like so.

     ______                 ______
.12|364.80     →    12|36480.

Now, do your division. Don't forget to bring your decimal point up to the quotient first. You still have to do that. So, you drop the decimal point from the divisor but not from the dividend, you always need that to bring it up to the quotient.

Just to give you a heads up, if the remainder is not zero, then that remainder will produce a fraction at the end of the quotient just like any division problem. You may think it strange to have a fraction in the decimal part of the answer, but that's how it works. You can get rid of that fraction (if you've a mind to) by adding zeros at the end of the dividend and continuing the cycle until you either get a remainder of zero or it is obvious that you never will. In the first case, you're done, you just got more digits in the answer (more precision) than you expected. In the second case, you'll never get a remainder of zero so your answer will be like for example .33333... forever. In that case you underline the repeating digits to show that they are infinitely repeating, and then you're done.

Now what? Of course. Go back to the chart and write in rules and examples. And of course throw it way.

In Conclusion

So, just what has your student, your child, your munchkin actually learned? She has learned to add, subtract, multiply, and divide whole numbers, fractions, mixed numbers, and decimals. She knows the rules, the steps, and she doesn’t get them mixed up. That’s a lot, wouldn’t you agree. How long would it have taken her to learn this in a public school? How many years? And how long did it take her here?

I hope you enjoyed my article, and you’re welcome.

PS: There are two additional things that could have been added to my chart and that I often do after they've mastered the material so far. And they are — First: One more column on the right for one more number form which is percents. Yes, you can add, subtract, multiply, and divide percents, but why would you want to? Well, your child will have to face percents sooner or later, but not until she's mastered the stuff already there. Then second: One more row on the bottom for one more operation which is compare. And this is important, not just on school tests but also on job application IQ tests. A question might be: "Arrange these numbers from smallest to largest: .2, ¹⁄₆, and 23%." I'm not going to show you how to do it, I'm just pointing out that comparison is an operation that sooner or later your child will have to confront. If you put it on my chart with the others, it won't be so baffeling because it too will have a place.