The power of 10

The questions in this activity are about multiplication and division by powers of 10. Powers of 10 are created by multiplying tens together. Some powers of 10 are:

• 10 x 10 = 100 (ten tens equal one hundred)
• 10 x 10 x 10 = 1 000 (ten times ten times ten equals one thousand)
• 10 x 10 x 10 x 10 = 10 000 (ten times ten times ten times ten equals ten thousand)
• 10 x 10 x 10 x 10 x 10 = 100 000 (ten times ten times ten times ten times ten equals one hundred thousand).

Powers of 10 can be written using exponents,

For example,

• 10³ = 1 000. The “3” indicates that 3 tens have been multiplied together. There are also 3 zeros in the product (1 000). Powers of 10 with an exponent of 1 or greater are counting numbers {1, 2, 3, 4, ...}. Powers of 10 may also be less than 1, but their meaning will be less obvious.

Students will find it helpful to create this pattern for exponents greater than or equal to 1 and then extend it to the left:

Powers of 10 are created by multiplication by 10, so moving one column to the left in the table above equates to division by 10.

For example,

• 1/10 of 100 is 10, so 1/10 of 1/10 is 1/100. Pages 22–27 of Book 7: Teaching Fractions, Decimals, and Percentages from the NDP resources describe how materials such as deci-mats or decimal pipes can be used to model these patterns in the place value system.

Students must be able to understand multiplication and division by powers of 10 if they are to handle more complex problems. Some may still think that “multiplication makes bigger” and “division makes smaller”. This overgeneralisation is based on what happens with whole numbers. As the examples in the chart below illustrate, the opposite is true in many cases. You need to work with your students to correct this common error of reasoning.

The questions in this activity use a measurement context for division rather than an equal-sharing context. Although it is reasonable to consider 12 objects measured in lots of 0.6, it is quite a stretch to imagine 12 objects shared into 0.6 equal sets. Problems that involve metric measurement are good vehicles for developing division by decimals.

Once the students have answered question 1, it is important that they discuss what they have found and can see the generalisation that their answer points to: If you measure the same amount in units that are 10 times smaller, the number of units that can be fitted in will be 10 times greater. The following two diagrams illustrate this mathematical principle.

12 ÷ 6 = 2 because 2 measures of 6 litres go into 12 litres.

12 ÷ 0.6 = 20 because 20 measures of 600 millilitres (0.6 litres) go into 12 litres. The measures are 10 times smaller, so 10 times as many measures fit into the whole (12 litres).

Similarly, if you multiply a number by a number that is 10 times greater than the one you used before, the result will be 10 times greater.

For example:

• 3 x 7 = 21, so 30 x 7 = 210. This means also that 0.3 x 7 = 2.1, because 0.3 is 10 times smaller than 3.

The parts of questions 3 and 5 can all be solved by multiplying or dividing the known result by 10 or a power of 10.

For example:

• 6 ÷ 3 = 2 has an answer that is 10 times smaller than the answer to 60 ÷ 3 = 20 or 10 times greater than the answer to 0.6 ÷ 3 = 0.2.

In some problems, both the starting number and the divisor have been changed by factors of 10, 100, 1 000, and so on.

For example:

• in question 3b, 2 ÷ 0.4 =can be easily solved using the known relationship 20 ÷ 4 = 5: the starting number 2 is 10 times smaller than 20, and 0.4 is 10 times smaller than 4, so the net effect is that the answer will be the same: 20 ÷ 4 = 5 and 2 ÷ 0.4 = 5.

In question 4, the students make generalisations about the effect of the size of the divisor on the results of divisions. Here are three very useful generalisations that apply to all positive numbers (things get more complicated when zero or negative numbers are involved):

• Dividing by a number greater than 1 will give a result that is less than the starting number. (For a ÷ b = c, if b > 1, then c < a.)
• Dividing a number by itself always gives 1. (For a ÷ b = c, if a = b, then c = 1.)
• Dividing by a number that is less than 1 but greater than 0 will give a result that is greater than the starting number. (For a ÷ b = c, if 0 < b < 1, then c > a.)

Two similar but different generalisations apply to multiplication of positive numbers:

• Multiplying by a number greater than 1 will give a result that is greater than the starting number. (For a x b = c, if b > 1, then c > a.)
• Multiplying by a number that is less than 1 but greater than 0 will give a result that is less than the starting number. (For a x b = c, if 0 < b < 1, then c < a.)

Encourage your students to discover these generalisations for themselves and to express them in their own words. They should generalise these properties across a range of numbers, including whole numbers and decimals. Using calculators can provide answers without the burden of calculation and allows the students to focus on the underpinning number relationships.

1.

a. 12 ÷ 0.6 = 20. (Ways of using the diagram  to show that 0.6 x 20 = 12 will vary.

For example:

• you might work out that there are 120 units in the diagram; 120 ÷ 6 = 20. Or you might see 12 lots of 0.6, plus 2 lots of 0.6 in the top 4 units of each set of 3 columns [there are 4 sets of 3 columns]: that’s 12 + (2 x 4) = 12 + 8 = 20.)

b. The answer to 12 ÷ 0.6 is 10 times bigger. (12 ÷ 0.6 = 20; 12 ÷ 6 = 2)

2.

a. Equations ii and iii

b. i. 10 times bigger. 12 ÷ 6 = 2; 0.2 x 10 = 2

ii. 100 times bigger. 12 ÷ 0.6 = 20; 0.2 x 100 = 20

iii. 1 000 times bigger. 12 ÷ 0.06 = 200; 0.2 x 1 000 = 200

iv. 10 times smaller. 12 ÷ 600 = 0.02; 0.2 ÷ 10 = 0.02

3.

a. 60 balloons. (18 ÷ 3 = 6, so 18 ÷ 0.3 = 60)

b. 5 pkts of cheerios. (20 ÷ 4 = 5, so 2 ÷ 0.4 = 5)

c. 25 kebab sticks. (700 ÷ 28 = 25, so 7 ÷ 0.28 = 25)

d. 0.4 kg or 400 g. (60 ÷ 15 = 4, so 6 ÷ 15 = 0.4)

4.

a. Answers will vary, but the divisor must be between 0 and 1,

For example,

• 14 ÷ 0.7 = 20, 16 ÷ 0.5 = 32, 12 ÷ 0.8 = 15.

b. Answers will vary, but the divisor must be 1,

For example,

• 14 ÷ 1 = 14, 202 ÷ 1 = 202, 88 ÷ 1 = 88.

5.

a. Multiply the answer to 32 ÷ 8 by 10 because 0.8 is of 8: 32 ÷ 8 = 4, so 32 ÷ 0.8 = 40.

b. Divide the answer to 32 ÷ 8 by 10 because 80 is 10 times bigger than 8: 32 ÷ 8 = 4, so 32 ÷ 80 = 0.4.

c. Divide the answer to 32 ÷ 8 by 100 because 800 is 100 times bigger than 8: 32 ÷ 8 = 4, so 32 ÷ 800 = 0.04.

d. Multiply the answer to 32 ÷ 8 by 100 because 0.08 is of 8: 32 ÷ 8 = 4, so 32 ÷ 0.08 = 400.

e. Divide the answer to 32 ÷ 8 by 10 because 3.2 is of 32: 32 ÷ 8 = 4, so 3.2 ÷ 8 = 0.4.

f. Divide the answer to 32 ÷ 8 by 100 because 3.2 x 10 = 32 and 8 x 10 = 80: 32 ÷ 8 = 4, so 3.2 ÷ 80 = 0.04. You could also do this in two parts: 3.2 is of 32, so divide the answer to 32 ÷ 8 by 10: so 3.2 ÷ 8 = 0.4. 80 is 10 times bigger than 8, so divide the answer to 3.2 ÷ 8 by 10: so 3.2 ÷ 80 = 0.04.