Place Value Properties

The number system used today throughout the world is known as a "Place Value System". It works very efficiently compared to other ancient systems such as Roman Numerals even though we have only 10 different "symbols" or digits -- 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 -- to work with. We can represent vast and tiny numbers simply by putting them in the right place or column.

For example, 21 904 is quite a large number, but, put a decimal point in it like this 21.904 -- and though the digits that make up the 2 numbers are identical, the second one is a thousand times smaller than the first.

The places or columns are labeled with powers of 10 and are numbered from right to left.

The first "place" is the "ones" column -- it indicates the multiplier of 10 ⁰ or 1. The second column indicates the multiplier of 10 ¹ or 10, and the third column indicates the multiplier of 10² or 100.

That's why we say "three hundred twenty-seven" when we see this number 327.
It means: (3 × 10²) + (2 × 10 ¹ ) + (7 × 10 ⁰ )

The power of 10 at the top of the column tells us how many zeros to put after the " 1 ".
The 4th place from the right then, is the 1000's column.
It tells us the multiplier of 10³ -- notice, there are 3 zeros.

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Adding: Carrying On

When we add small numbers -- say 3 and 4 -- together, we don't have to pay attention to the column or place in which to write the sum, because it is smaller than 10. We write it in the one's column. But, if our sum is greater than 10 -- say we have 7 + 9 -- the answer is 16 -- that's 1-ten and 6-ones. Since we're only allowed to put one digit in each column -- we can't write 16 in the 1's column -- so we carry one 10 to its rightful place -- the 10's column and write 16.

Though it is called Regrouping in many text books, this operation should be called CARRYING -- because we carry a ten over a column, to its proper place. We do the same with 100's, 1000's and gazillions of course. (There's no units called gazillions).

Example

Now we add with 2 different methods: traditional and column sum

Note: We have to line things up properly with the Column Sum method.

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Subtraction: Borrowing from a Neighbor

When we subtract a small number from a big one, say 9 - 4, we write 5 in the 1's column and we're done. However, say we have to find 65 – 37. Now, we have to subtract 7 from 5 in the one's column, but we can't do it, because 7 is bigger than 5. So, we borrow 10-ones from the neighboring number -- the 6 -- in the ten's column. We think of 65 as 5 tens + 15 ones -- and now we can subtract 7 from 15 in the 1's column, to get 8. The difference in the 10's column now is 50 – 30 or 20. The answer is 28.

In order to check our subtraction, we add 28 to 37 to get 65.

Long ago, in the days before the check out cash was computerized, cashiers had to calculate the correct change to give shoppers when they paid. They did it with addition. After all, subtraction is just backward addition. If a customer paid for $2.85 worth of stuff with a $5 bill, the cashier would figure out the change by adding from $2.85 up to $5. She'd think: two-eighty-five plus 15¢ makes 3 dollars plus 2 more dollars makes $5. The correct change is 2 dollars and 15 cents.

2.85 + 0.15 = 3.00 + 2.00 = 5.00

Just as we can add in 2 different ways, now we know we can subtract by adding.

Example

No matter how big or small the numbers we're adding or subtracting, the approach is the same. When we carry or borrow from the 100's, 1000's or gazillion's column -- we always carry or borrow 10 of those units from or to the column on the right.

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Estimating the Sum or Difference: making an educated guess

Tanya is going to the market for her mom. Her shopping list has five items on it. There's a jar of peanut butter for 99¢, a raisin bread for $1.89, eight small yogurt cups for $5.59, a tub of margarine that costs $1.29, and 5 lbs. of potatoes for $2.87. Her mom wants to know how much money to give her for the groceries, so together they estimate the total, by rounding each price up or down, and then adding their estimates.

They use the rules for rounding that say:

round up if the number is at halfway or more,
smaller than halfway, round down.

Tanya and her mom make these estimates for the shopping list items:

$1.00 + $2.00 + $6.00 + $1.00 + $3.00 = $13.00.

So, Tanya's mom gives her $15.00 and sends her on her way. At the checkout, Tanya has to pay $12.63 for her order -- just 37¢ less than the estimate of $13.00 she and her mom made.

When to Break the Rounding Rules

Sometimes, instead of rounding according to the rules, we have to use common sense. For example, if we have 247 – 192, and we round to the hundred's place, we would get 200 – 200, so the difference is = 0. In a case like this, it is best to round to the ten's place. Here, our estimate would be 250 – 190 = 60. The actual difference is 55, which is a lot closer to 60 than it is to 0.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

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Practice Exercise 1: Addition

1) Estimate the sum, then use the COLUMN SUM METHOD to find:

a)

b)

c)

d)

2) Use the TRADITIONAL METHOD with CARRYING to find these sums:

a)

b)

c)

d)

Practice Exercise 2: Subtraction

1) Estimate the difference, then use the CASHIER'S METHOD to find:

a)

b)

c)

d)

2) Use the TRADITIONAL METHOD with BORROWING to find these differences:

a)

b)

c)

d)

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Solutions

Practice Exercise 1: Addition

1) Estimate the sum, then use the COLUMN SUM METHOD to find:

a) ......4 + 9 = 13 .....30 + 80 = 110 ..200 + 100 = 300 ..234 + 189 = 423 est: 200 + 200 = 400

b) ......6 + 7 = 13 .....70 + 90 = 160 ..500 + 200 = 700 ..576 + 297 = 873 est: 600 + 300 = 900

c) ......2 + 9 = 11 .....50 + 90 = 140 ....400. + 0 = 400 ....452 + 99 = 551 est: 450 + 100 = 550

d) .......7 + 6 = 13 ........30 + 60 = 90 .700 + 300 = 1000 ..737 + 366 = 1103 est: 700 + 400 = 1100

2) Use the TRADITIONAL METHOD with CARRYING to find these sums:

a) 494 + 37 = 531

b) 397 + 253 = 650

c) 178 + 544 = 722

d) 779 + 69 = 848

Practice Exercise 2: Subtraction

1) Estimate the difference, then use the CASHIER'S METHOD to find:

a) 189 + 11 = 200 200 + 34 = 234 11 + 34 = 45 est: 230 – 190 = 40

b) 297 + 3 = 300 300 + 276 = 576 3 + 276 = 279 est: 600 – 300 = 300

c) 99 + 1 = 100 100 + 352 = 452 1 + 352 = 353 est: 450 – 100 = 350

d) 366 + 34 = 400 400 + 337 = 737 34 + 337 = 371 est: 700 – 400 = 300

2) Use the TRADITIONAL METHOD with BORROWING to find these differences:

a)

b)

c)

d)

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