B2.3 Use mental math strategies to multiply and divide whole numbers and decimal numbers up to thousandths by powers of ten, and explain the strategies used.
Skill: Using Mental Math Strategies to Multiply and Divide Whole Numbers and Decimal Numbers to Thousandths by Powers of 10, and Explain the Strategies Used
Mental Math
Daily life presents many opportunities to perform operations with decimal numbers. For example, shopping and measuring involve decimal numbers. Estimation skills and mental math skills are characteristics of number sense and operations sense. A variety of mental math strategies can be used, including rounding, decomposition, and the use of benchmarks.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 91.
Mentally multiplying and dividing whole numbers and decimals by powers of ten builds on the constant 10:1 ratio that exists between place-value columns.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Junior students already have experience with the concept of multiplication and understand that multiplying whole numbers results in a product that is greater than the quantities involved, except in situations involving multiplication by 1 and by 0. They are then often surprised to find that a multiplication in which one of the factors is a decimal number less than 1 gives a product less than the other factor (for example, \(10\; \times \;0,1\; = \;1\;{\rm{et}}\;{\rm{20}}\; \times \;{\rm{0}}{\rm{,10}}\;{\rm{ = } }\;{\rm{2}}\)).
These results are surprising to anyone who does not deal with numbers in context or has not developed an understanding of multiplication to explain such results.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 79.
When you multiply a whole number by 0.1 the value becomes one tenth as large, which is the same as dividing it by 10.
When you multiply a whole number by 0.01, the value becomes one one hundredth as large, which is the same as dividing it by 100.
When you multiply a whole number by 0.001, the value becomes one one thousandth as large, which is the same as dividing it by 1000.
Multiplication Performed Using the Place Value Chart
\(23 \times 0.1 = 2.3\)
I visualize a shift of 1 column to the right in the place value chart.
\(23 \times 0.01 = 0.23\)
I visualize a shift of two columns to the right in the place value chart.
Multiplication Using Division
\(23 \times 0.1 = 2.3\)
To perform \(23\; \times \;0.1\;\), I visualize the number 23 divided into 10 equal groups. I decompose 23 into 20 + 3.
\(\begin{array}{l}20\; \div \;10\; = \;2\\3\; \div \;10\; = \;0,3\\2\; + \; 0.3\;=\;2.3\end{array}\)
So, in each group of 10, there are 2.3.
\[230\; \times\;0.01\; = \;2.3\]
To perform \(230; \times; 0.01; \), I visualize the number 230 divided into 100 equal groups. I decompose 230 into 200 + 30.
So in each of the groups of 100, there are 2.3.
Source : adapted from En avant, les maths!, 5e année, ML, Nombres, p. 5-6.
When a whole number or decimal number is multiplied by a positive power of 10, the value becomes larger (shifts one position) for each multiplication by 10.
- \( \times \;{10^1}\): moving one position to the left
- \( \times \;{10^2}\): a move of two positions to the left
- \( \times \;{10^3}\): a move of three positions to the left
When dividing a whole number or a decimal number by a positive power of 10, the value becomes smaller (moving one position) for each multiplication by 10.
- ÷ 101 : moving one position to the right (× 0.1)
- ÷ 102 : a move two positions to the right (× 0.01)
- ÷ 103 : a move three positions to the right (× 0.001)
This is evident when using a place value chart.
This way, the student can write their initial number in the right place on the chart and continue with their mental math by moving the numbers according to the positions. This visual would make it easier for them to apply the mental math strategies and explain them.
Example
Initial number: 645.123
- 645.123 × 101
- 645.123 × 102
- 645.123 × 103
- 645.123 ×104
- 645.123 ÷ 101 or 645.123 ×0.1
- 645.123 ÷ 102 or 645.123 ×0.01
- 645.123 ÷ 103 or 645.123 × 0.001
| Millions | Hundreds of Thousand | Tens of Thousand | Thousands | Hundreds | Tens | Units. | Tenths | Hundredths | Thousandths | |
|---|---|---|---|---|---|---|---|---|---|---|
| 6 | 4 | 5, | 1 | 2 | 3 | |||||
| a) | 6 | 4 | 5 | 1, | 2 | 3 | ||||
| b) | 6 | 4 | 5 | 1 | 2, | 3 | ||||
| c) | 6 | 4 | 5 | 1 | 2 | 3, | ||||
| d) | 6 | 4 | 5 | 1 | 2 | 3 | 0, | |||
| e) | 6 | 4, | 5 | 1 | 2 | |||||
| f) | 6, | 4 | 5 | 1 | ||||||
| g) | 0, | 6 | 4 | 5 |
What do you notice if I multiply the initial number by 1,000? By 10,000? By 0.01? By 0.001?
Knowledge: Place Value Relationship
The beginning of the junior division is the first time students study the decimal part of a number. They need to deepen their understanding of the place values of the numbers and the relationship between the place values. Decimal numbers are part of everyday life and understanding the place values to the right and left of the decimal point is essential. The decimal point plays a significant role in decimal notation. It separates the whole part from the decimal part and thus indicates the position of the units.
It is essential that students recognize the position of the units, since it is the unit that defines the whole, from which the tenths, hundredths and thousandths are formed on the one hand, and the tens, hundreds and thousands on the other. We can therefore say that the unit, identified by the point, is at the heart of the decimal system.
This recognition of the role of the units digit is evidenced by the name prefixes given to the place value of the digits on either side of the unit digit. Thus, tens represent a quantity ten times greater than the unit, while tenths represent a quantity ten times smaller than the unit.
Note: Some students are under the impression that the decimal point is the center of the decimal system. As a result, they tend to call the first position to the right of the decimal point the oneth rather than the tenth position. It is important that students also understand the multiplicative relationship by 10 that exists between adjacent place values. They have previously developed an understanding of this relationship in the study of whole numbers, that each place has a value 10 times greater than the one to its right and 10 times smaller than the one to its left.
This multiplicative relationship is true for all positions.
Students can develop an understanding of this by doing groupings using base ten materials. The idea is to demonstrate that just as 10 ones yield 1 ten, 10 tenths yield 1 one and 10 hundredths yield 1 tenth, and so on.
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 51-52.