C1.4 Create and describe patterns to illustrate relationships among whole numbers and decimal numbers.
Skill: Creating and Describing Number Patterns to Represent Relationships Between Whole Numbers and Decimal Numbers
The base ten system includes multiple patterns that help deepen understanding of number relationships.
Patterns can be used to understand the relationships between whole numbers and decimal numbers.
Source: translated from Curriculum de l'Ontario, Programme de mathématiques de la 1re à la 8e année, 2020, Ministère de l'Éducation de l'Ontario.
Understanding the relationships between the value of digits and their position in a number is essential to the development of number sense. In the Primary Division, students develop an understanding of the relationships between the place values of ones, tens and hundreds. However, in the Junior Division, they do not automatically transpose this understanding to larger numbers. This is why teachers must be sure that students understand that the value of any position in a number is always 10 times greater than the value of the position immediately to the right, and 10 times less than the value of the position immediately to the left. It is also important to examine the relationships (for example, 100 times or 1,000 times greater or less than) between place values in order to develop students' deep number sense, including sense of large numbers.
Students should also recognize, for example, that ten thousand represents a grouping of 100 hundreds, a grouping of 1,000 tens, or even a grouping of 10,000 ones. These groupings allow students to recognize equivalent representations of numbers (for example, 2 534 is equal to 25 hundreds and 34 ones).
Example
The number of ones increases by 10 when the number of tens decreases by 1.
| Number | Tens | Ones |
|---|---|---|
| 56 | 5 | 6 |
| 56 | 4 | 16 |
| 56 | 3 | 26 |
| 56 | 2 | 36 |
| 56 | 1 | 46 |
| 56 | 0 | 56 |
Place value relationships play an important role when doing estimation, rounding, or decomposition. In addition, they are the basis for multiplication and division by a multiple of 10.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 44-45.
Example
When the factor that is multiplied by 10 increases by 1, the product increases by 10.
When the dividend decreases by 10, the quotient decreases by 1.
| × 10 | ÷ 10 |
|---|---|
| 1 × 10 = 10 | 100 ÷ 10 = 10 |
| 2 × 10 = 20 | 90 ÷ 10 = 9 |
| 3 × 10 = 30 | 80 ÷ 10 = 8 |
| 4 × 10 = 40 | 70 ÷ 10 = 7 |
| 5 × 10 = 50 | 60 ÷ 10 = 6 |
| 6 × 10 = 60 | 50 ÷ 10 = 5 |
| 7 × 10 = 70 | 40 ÷ 10 = 4 |
| 8 × 10 = 80 | 30 ÷ 10 = 3 |
| 9 × 10 = 90 | 20 ÷ 10 = 2 |
| 10 × 10 = 100 | 10 ÷ 10 = 1 |
Place Value Relationships in Decimal Numbers
The beginning of the junior grades is the first time students study decimal numbers. Students need to deepen their understanding of the place value of numbers and the relationships between 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.
It is important that students also grasp the multiplicative relationship by 10 that exists between adjacent position values. Students have previously developed an understanding of this relationship in the study of natural numbers, that each position 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 also true for decimal 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.
Example
The number of thousandths increases by 10 when the number of hundredths decreases by 1.
| Decimal Number | Tens | Ones | Tenth | Hundredth | Thousandth |
|---|---|---|---|---|---|
| 56.472 | 5 | 6 | 4 | 7 | 2 |
| 56.472 | 5 | 6 | 4 | 6 | 12 |
| 56.472 | 5 | 6 | 4 | 5 | 22 |
| 56.472 | 5 | 6 | 4 | 4 | 32 |
| 56.472 | 5 | 6 | 4 | 3 | 42 |
| 56.472 | 5 | 6 | 4 | 2 | 52 |
| 56.472 | 5 | 6 | 4 | 1 | 62 |
| 56.472 | 5 | 6 | 4 | 0 | 72 |
The student can create patterns of related operations to illustrate the relationship between addition and subtraction facts.
Example
When the first term in the addition increases by 0.001 and the second term decreases by 0.001, the sum is always the same throughout the series of related operations.
When the second term in the subtraction decreases by 0.001, the difference increases by 0.001 throughout the series of related operations.
| \(\ 74.000 + 0.008 = 74.008\) |
\(\ 74.008 – 0.008 = 74.000 \) |
| \(\ 74.001 + 0.007 = 74.008 \) | \(\ 74.008 – 0.007 = 74.001 \) |
| \(\ 74.002 + 0.006 = 74.008\) | \(\ 74.008 – 0.006 = 74.002\) |
| \(\ 74.003 + 0.005 = 74.008\) | \(\ 74.008 – 0.005 = 74.003 \) |
| \(\ 74.004 + 0.004 = 74.008\) | \(\ 74.008 – 0.004 = 74.004\) |
| \(\ 74.005 + 0.003 = 74.008\) | \(\ 74.008 – 0.003 = 74.005 \) |
| \(\ 74.006 + 0.002 = 74.008\) | \(\ 74.008 – 0.002 = 74.006\) |
| \(\ 74.007 + 0.001 = 74.008\) | \(\ 74.008 – 0.001 = 74.007 \) |
| \(\ 74.008 + 0.000 = 74.008\) | \(\ 74.008 – 0.000 = 74.008\) |
The multiplicative relationship by 10 between positional values can also be explored with metric units of measurement, since these are designed in terms of the base ten (for example, 10 cm yields 1 dm, 10 mm yields 1 cm). Monetary units are best avoided in early conceptual learning, because while there is indeed a multiplicative relationship by 10 between the value of some coins (for example, between pennies and dimes or between dimes and $1s), it is quite different between other coins (for example, between nickels and $1s).
The calculator offers interesting ways to explore the multiplicative relationship by 10. For example, students can perform the operation 0.1 + 0.1, predict the answer, and continue the series until they arrive at 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1. Before adding 0.1 one last time, a discussion is needed to explore students' assumptions and reasoning, and to ensure that they recognize that 10 tenths is 1 unit. Students can repeat the activity with the series 0.01 + 0.01 +… or 0.001 + 0.001 +…or with other decimal numbers.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Modélisation et algèbre, p. 51-53.