Multiplying by a number that is less than 1 but greater than 0 will give a result that is less than the starting number.Multiplying by a number greater than 1 will give a result that is greater than the starting number.Two similar but different generalisations apply to multiplication of positive numbers: Dividing by a number greater than 1 will give a result that is less than the starting number. Here are three very useful generalisations that apply to all positive numbers (things get more complicated when zero or negative numbers are involved): In question 4, the students make generalisations about the effect of the size of the divisor on the results of divisions. 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 some problems, both the starting number and the divisor have been changed by factors of 10, 100, 1 000, and so on. 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. 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. This means also that 0.3 x 7 = 2.1, because 0.3 is 10 times smaller than 3. 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. The measures are 10 times smaller, so 10 times as many measures fit into the whole (12 litres). The following two diagrams illustrate this mathematical principle.ġ2 ÷ 6 = 2 because 2 measures of 6 litres go into 12 litres.ġ2 ÷ 0.6 = 20 because 20 measures of 600 millilitres (0.6 litres) go into 12 litres. 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. Problems that involve metric measurement are good vehicles for developing division by decimals. 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. The questions in this activity use a measurement context for division rather than an equal-sharing context. You need to work with your students to correct this common error of reasoning. As the examples in the chart below illustrate, the opposite is true in many cases. This overgeneralisation is based on what happens with whole numbers. Some may still think that “multiplication makes bigger” and “division makes smaller”. Students must be able to understand multiplication and division by powers of 10 if they are to handle more complex problems. 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. 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. 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 may also be less than 1, but their meaning will be less obvious. Powers of 10 with an exponent of 1 or greater are counting numbers. There are also 3 zeros in the product (1 000). The “3” indicates that 3 tens have been multiplied together. Powers of 10 can be written using exponents, for example, 10 3 = 1 000. Some powers of 10 are:ġ0 x 10 = 100 (ten tens equal one hundred)ġ0 x 10 x 10 = 1 000 (ten times ten times ten equals one thousand)ġ0 x 10 x 10 x 10 = 10 000 (ten times ten times ten times ten equals ten thousand)ġ0 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 10Īre created by multiplying tens together. The questions in this activity are about multiplication and division by powers of 10.
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