The power of powers

Introduce these activities to an independent group by reading what Aarif and his teacher say about 103. Reinforce that 10 to the power of 3 means 10 x 10 x 10, not 10 x 3 or 3 tens.

Encourage the students in a guided teaching group to say “I moved the 1 along 8 places” rather than “I added 8 zeroes” when they are predicting what 10⁸ would be. The rule “add a zero” to multiply by 10 is conceptually inaccurate. Mathematically, adding 0 to a number doesn’t change it: 12 + 0 = 12. This “rule” also sets students up for problems later when they are multiplying decimal fractions because 5.6 x 10 does not equal 5.60. It’s better for students to learn to understand mathematical principles that are always true than to learn lots of rules that need to be continually changed in different situations.

The generalisation in this case is that when we multiply by 10, the numbers shift along 1 place to the left; it is the digits that move, not the decimal point. You can demonstrate this by writing the number 43 on a strip of paper under the place value houses and moving the strip along 1 place to the left to multiply by 10. This creates a space under the ones column, where a 0 has to be written as a “place-holder” because if we didn’t have the house labels above the numbers, we wouldn’t know whether the number should be 43 or 430. It is important to use place value materials to support this generalisation.

Typically, students will jump to the conclusion that 100 is 0, but it isn’t! The easiest way to demonstrate this is to use a table or place value houses and get the students to follow the pattern from the left to the right, starting at 10⁵ and dividing by 10 each time to reduce the power. Make sure that when they get to 10², they see it as 100. So 10¹ is 100 ÷ 10 = 10, and 100 is 10 ÷ 10 = 1.

For your own information, note that anything to the power of 0 (apart from 0 itself) has the value of 1. For example,

• in the extension activity for page 5, Megabytes of Memory, the pattern leads to 2⁰, which is 2 ÷ 2 = 1.

Place value houses are an effective way of helping students to say large numbers. Encourage them to say the number in each house and then the name of the house. For example,

• 483 065 000 would be said “four hundred and eighty-three million, sixty-five thousand”.

In question 2, students work with powers of numbers that are themselves multiples of 10. They are likely to find it helpful to rewrite each number using only 10s. In this way, they will see that:

• 1003 = 100 x 100 x 100
• = (10 x 10) x (10 x 10) x (10 x 10)
• = 10 x 10 x 10 x 10 x 10 x 10
• = 106

A key idea is that the grouping of factors does not affect the value:

• 1003 = (10 x 10) x (10 x 10) x (10 x 10)
• = (10 x 10 x 10) x (10 x 10 x 10)
• = (10 x 10) x (10 x 10 x 10 x 10)
• = 10 x 10 x 10 x 10 x 10 x 10

For questions 3 and 4, some students might find it helpful to manipulate or image tens money ($100, $10, $1) and work out how many tens and hundreds are in numbers by making exchanges. For example,

• they may work out that 103 is $1,000, which is the same as ten $100 notes or one hundred $10 notes.

Other students may be able to manipulate the numbers in question 3 to work out how many tens are in a number by partitioning one lot of 10 out of the factors. 103 can be written as (10 x 10) x 10, which is the same as 100 x 10, so there are 100 lots of 10 in 103.

To work out how many hundreds are in the numbers in question 4, some students may be able to partition 100, or 10 x 10, out of the list of factors.

For example,

• 1002 can be written as (10 x 10) x (10 x 10) = 10 x (10 x 10 x 10)  = 10 x 1 000 (ten thousand)

or

• (10 x 10) x (10 x 10) = 100 x 100 = 10 000

Activity 1

1.

2.

a.

• i. 100 000 000
• ii. 10
• iii. 1

b. Answers will vary, but the basic idea is that every time you multiply by 10, the 1 moves along one place to the left. As the power gets bigger, you multiply by more lots of 10 and move the 1 along more places. When you are reducing the power by 1, for example, from 10³ to 10², you are dividing by 10. 10² is 100, so 10¹ is 100 ÷ 10 = 10. 10 ÷ 10 = 1, so 100 is 1.

Activity 2

1.

a. On your place value houses, you should have these powers of 10:

b.

• i. 10⁶: one million
• ii. 10¹¹: one hundred billion
• iii. 10⁹: one billion
• iv. 10⁰: one

2.

a.

• i. 1 000 000 (100 x 100 x 100)
• ii. 1 000 000 (1 000 x 1 000)
• iii. 100 000 000 (10 000 x 10 000)
• iv. 100 000 000 (100 x 100 x 100 x 100)

b. 100 can be written as 10 x 10, so 100³ can be written as (10 x 10) x (10 x 10) x (10 x 10), which is 10⁶ or 1 000 000.  1 000 can be written as 10 x 10 x 10, so 1 000² can be written as (10 x 10 x 10) x (10 x 10 x 10), which is also 10⁶ or 1 000 000. A similar process can be used to show that 10 000² = 100⁴.

c.

• i. A billion (1 000 000 000)
• ii. A trillion (1 000 000 000 000)
• iii. A hundred million (100 000 000)

3.

a. 100 tens

b. 100 000 tens

c. 1 ten

d. 1 000 tens

e. 1 000 tens

f. 100 000 tens

4.

a. 10 hundreds

b. 100 hundreds

c. 10 000 hundreds

d. 10 000 hundreds

e. 10 000 hundre