Tens time

This activity explores what happens to whole numbers when they are divided by 10 and 100. Students will need some understanding of the meaning of the digits in one- and two-place decimals.

This is a major concept for students in their understanding of our number system and how they operate it. It will probably take a number of lessons before most students are confident with the workings of the decimal system. (The Number booklet also deals with decimals on pages 13, 16, and 17.)

The following concepts may have to be revisited before this activity is attempted:

• What happens to numbers when they are multiplied by 10 and 100.
• Each column in our number system grows 10 times larger as we move to the left.
• Each column in our number system grows 10 times smaller as we move to the right.
• The meaning of fractional notation, for example, 3/10 means three parts out of the 10 that make up the unit.
• The values of digits in the first and second decimal places.
• The role of the decimal point in establishing the ones or units column immediately to its left.

Patterning can be a valuable way of helping students grasp the ideas involved in this activity. Usethe constant function on a simple calculator to show the patterns of multiplying and dividing by 10. This can be very effective in helping students understand the concept of decimal places. To set up multiplying by 10 as a constant function, they will have to press the 10 x keys first. (Some calculators may require 10 x x keys to be pressed.)

After they have entered 10 x , ask students to enter 6 on their calculator. Ask them:

• What will the digits be if we multiply this by 10? (60 or six zero) We can answer this by simply pressing the = key. This will multiply 6 by 10, and the display will now show 60. Watch what happens to the 6 as we continually multiply by 10. (60 becomes 600, 600 becomes 6 000, 6 000 becomes …)

The students should be able to see the pattern. You may want the class to verbalise this:

• As the 6 moves from tens to hundreds to thousands and so on, it moves one place to the left.

To show dividing by 10 on the calculator using the constant function, the students could enter 600 000 ÷ 10 = = = = . Each time they press = , the calculator will divide by 10.

(On some calculators, students may need to enter 10 ÷ ÷ 600 000 = = = = .)

Have the students press = until the 6 has moved down to the ones place. Discuss what will happen if they press = again. Discuss the meaning of the display 0.6 as six-tenths and how the 6 in this place is 10 times less than the 6 was in the ones place. Have them press = again to show 0.06. Discuss the meaning of this as well.

You may wish to model this with place value blocks. Emphasise the use of the decimal point to create the ones place as shown in the following diagram:

This activity is a good context for applying the students’ understanding of the place value when a number is divided by 10.

After students have calculated the share value as $86.40, take the opportunity to compare the language of “eighty-six dollars and forty cents” with the decimal equivalent, “eighty-six point four dollars.” Students may not realise that these are equivalent statements. Make sure students can explain why 40 cents is the same as 0.4 of a dollar.

Activity 1

1.

He can think of the 5 as being 50 tenths and then divide the 50 tenths by 10. The answer is 5 tenths, which is 0.5 (1/2).

2.

a. 43.5

b. Answers will vary but could include: I used Mike’s method and then added 40 + 3 + 0.5 to get 43.5.

3.

a. 76

b. 27.5

c. 6.4

d. 70.3

e. 5.94

f. 3.5

g. 56

h. 0.8

Activity 2

$86.40

Discussion will vary.