Prologue 1

ELEMENTARY ADDITION

Sums less than 10

Sums equal to 10

Sums between 10 and 20

Doubling

Zero

Practice problems

ARITHMETIC, as we point out in the Introduction, is and always has been a spoken skill, based on counting. That skill begins with elementary addition, the subject of this page

The student should not get conditioned to having to write
and see such simple sums as 5 + 2.

In fact, the student is encouraged to write as little as possible.

The counting-numbers are 1, 2, 3, 4, and so on. The problem of addition, of "adding" any two of them --

6 + 3 = ?

-- is the problem of naming the number that results when, starting with the first, you continue counting as many times as there are 1's in the other.

Since 3 is three 1's -- 3 = 1 + 1 + 1 -- then, starting at 6, continue counting three times.

"Six -- seven, eight, nine."

6 + 3 = 9.

The addition sign is this:  + , which we read "plus."  We call the answer to an addition problem the sum; we call the numbers we are adding the terms of the sum.  And so when we add 6 + 3, the terms are 6 and 3; their sum is 9.  We also call '6 + 3' a sum -- even if we do not name the answer.

Skill in adding begins with remembering that 6 + 3 is 9 -- without having to count it. The student should begin by knowing all the sums that are less than 10.

3 + 2 = 5,  4 + 3 = 7,  5 + 3 = 8,

and so on.

In fact, once you know that

You will know how to count by 10's. That is extremely important because our system of numbers is based on 10's.

Problem 1.   Starting with 10, count by 10's up to 100.

To see the answer, pass your mouse over the colored area.
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10, 20, 30, 40, 50, 60, 70, 80, 90, 100.

Those numbers are called the multiples of 10.

Problem 2.   Starting with 24, count by 10's up to 104.

24, 34, 44, 54, 64, 74, 84, 94, 104.

Problem 3.   2 + 4 = 6.   20 + 40= 60.   24 + 40 = 64.

Problem 4.   5 + 3 = 8.   50 + 30= 80.   52 + 30 = 82.

Problem 5.   7 + 2 = 9.   70 + 20= 90.   75 + 20 = 95.

The order property.  An elementary fact of addition is that the order in which we add two numbers does not matter.  If you know that

6 + 3 = 9,

then you would also know that

3 + 6 = 9.

But it is simpler to add 3 to 6, the smaller to the larger, than it is to add 6 to 3, the larger to the smaller.

In any case, the order does not matter.  That will be true for any number of terms.

2 + 3 + 4 = 3 + 2 + 4 = 4 + 3 + 2 = 9.

Now practice these sums that are less than 10.

Sums equal to 10

Next, it is useful to know all the ways of adding two numbers to make 10.

Sums between 10 and 20

Finally, the student should know sums such as  9 + 6 = 15 ,  8 + 5 = 13, and so on. To become familiar with them, you can first make 10 by regrouping.  But with repetition you must know them.

Example 1.   9 + 6.

Make 10 by taking 1 from 6, leaving 5:

9 + 6 = 9 + 1+ 5 = 10 + 5 = 15.

Say, "9 + 1 is 10, plus 5 is 15."

That is, regroup 1 with 9 to make 10.

Example 2.   5 + 7.

"5 + 5 is 10, plus 2 is 12."

Example 3.     3 + 8.

"8 + 2 is 10, plus 1 is 11."

Problem 6.   9 + 5 = 14.   90 + 50= 140.   92 + 50 = 142.

We will continue this in Lesson 5.

Doubling

It can help to know the sum of a number added to itself.

5 + 5 = 10

6 + 6 = 12

7 + 7 = 14

8 + 8 = 16

9 + 9 = 18

For, once you know that

6 + 6 = 12,

then you could know that

6 + 7 = 13,

and

6 + 5 = 11.

7 + 7 = 14

7 + 8 = 15

7 + 6 = 13

8 + 8 = 16

8 + 9 = 17

8 + 7 = 15

9 + 9 = 18

9 + 8 = 17

Zero

0 is a number. It can answer the question, "How many?" We will see that when we write a number like 502, the 2 tells us how many ones; the 0 tells us how many tens; and the 5, how many hundreds.

We can say that 0 is that number such that when you add it to any number, that number does not change.

5 + 0 = 5.

0 + 6 = 6.

0 means no units of whatever kind.

There is a subtle difference between 0 units and nothing. Say that you have an account at the First National Bank and that your balance is $10. If you now withdraw $10, your balance is 0 dollars. But if you do not have an account at that bank, then you do not have a balance of 0 dollars. You have nothing!

That is the difference between 0 and nothing.

*

Practice the following until you remember each one.

The multiplication table

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