Teachers' Guidebook for Primary Mathematics

Excerpt from a guidebook in Japan

Division(2) 6 hours

2 hours are assigned to the following two pages.

The charge of 34 pieces of colored papar was 714 yen.
How much is a piece ?

[1] What place number is 714

34 ?
2 place number

Becase 34

71, the quotient is a 2 place number.

When you divide a number by a 2 place number, see which of the divisor and the 2 place number consisting of first two numbers of the divident is bigger, and see what place number the quoetient is.

[2] Take 70 and 30 instead of 71 and 34, respectively. And guess what place number the quotient is.

p.49

The goal of this subunit
1. Understanding a way of calculation of (3 or 4 place number)

(2 or 3 place number)

The goal of these two hours instruction
Understanding a way of calculation of (3 place number)

(2 place number)

Example of Process

Asking question 1
(Reading the problem) "Let's make a numeral expression of the problem, and consider the way of calculation."
Orientation
Letting students guess what the quotient is by considering 71

34, where "71" is the first two numbers of the devident "714".
Letting students understand that the place where the first number of the quotient comes is the ten because 34

71.
Letting students recall how they did calulations in the previous lessons, and summing up the way of determining the place where the first number of the quotient comes.

Asking question 2
Moving to Question 2.
Orientation
Letting students notice the followings:
By the reason of
71

2
"2" comes to the tens.
The operation of "lowering 4 which is placed in the units" is needed in order that to get the partial quotient which comes to the place of the units.

[3] Let us sum up the way of calculation.

How to Calculate 71434

(1)
(2)	Here are 864 apples. If you divide them equally among 24 boxes, how many apples are there in each box ? Expression: 86424 = 36, Answer: 36 apples
(3)	If you cut a 4m50cm length string into pieces each of which is 25cm length, how many pieces you get ? Expression: 45025 = 18, Answer: 18 pieces

p.50

Asking question 3
"Let us sum up the way of calculation of 714

34."
Orientation
Letting students understand the followings:

1.	If the place where the first number of the quotient comes is just above the n-th place of the divident-as-a-sequence-of-numbers, then n is the minimum of such number k as the number expressed by the first k numbers of the divident is bigger than the divider.
2.	The way of calculation in the case where the divider is a 2 place number is the same as the case where the divider is a 1 place number. That is, "Place Multiple Subtract Lower".
3.	"3" resulted from "71-68" is actually 30. Lowering "4" means adding 4 to 30.

[1] Exercise of the calculation of (3 place number)

(2 place number) = (2 place number)

[1] Exercise of applying to real problems the calculation of (3 place number)

(2 place number) = (2 place number)
(Getting-The-Multiplicand type problems)

[1] Exercise of applying to real problems the calculation of (3 place number)

(2 place number) = (2 place number)
(Getting-The-Multiplier type problems)

On the algorithm of the division The way of division is founded on the decimal numeration system. Thus, teachers should instruct students not only the calculation techniques but also the reason of this way of calculation, that is, its structure, principle and usefulness.
The algorithm of the division is as follows :
1. Determining the place where the first number of the quotient comes, and getting the partial quotient.
2. Getting the partial product. (multiplying)
3. Subtracting the partial product from the dividend. Checking whether the difference is smaller than the divider or not. (subtraction)
4. Lower the next-to-come number of the dividend.
The goal of this instruction is that students understand this algorithm and that they can confirm the result of calculation by using the formula : (divider) (quotient) + (remainder) = (dividend).