C1.4 Create and describe patterns to illustrate relationships among whole numbers up to 1000.
Skill: Creating and Representing Number Patterns Involving Whole Numbers up to 1000
There are many patterns within the whole number system that help deepen understanding of number relationships.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8, 2020, Ontario Ministry of Education.
Students demonstrate their understanding of the concept of patterns by creating and explaining number patterns (e.g., decompositions of numbers, sets of related operations). This is an opportunity for students to make connections between addition and subtraction and between multiplication and division, to understand the inverse relationship between these related operations, and to use these connections in calculations with larger numbers.
Teachers can have students create a pattern with operations using larger numbers. Students can then construct the related pattern with the associated subtraction facts and apply their knowledge when doing calculations with larger numbers.
In this way, students notice that addition and subtraction are inverse operations even when the numbers are larger.
Example
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604 |
|
|
\(600 + 4 = 604\) |
\(604 - 4 = 600\) |
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\(601 + 3 = 604\) |
\(604 - 3 = 601\) |
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\(602 + 2 = 604\) |
\(604 - 2 = 602\) |
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\(603 + 1 = 604\) |
\(604 - 1 = 603\) |
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\(604 + 0 = 604\) |
\(604 - 0 = 604\) |
This allows students to establish the inverse relationship between addition and subtraction when doing calculations with larger numbers.
Teachers can also have students create a pattern of operations using concrete or semi-concrete materials (e.g., nesting cubes, base 10 materials). Students can then construct the related series with associated division facts and describe it.
Example
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\(\times\) |
\(\div\) |
|---|---|
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\(\ 7 \times 1\ = 7\) |
\(\ 7 \div 1\ = 7\) |
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\(\ 7 \times 2\ = 14\) |
\(\ 14 \div 2\ = 7\) |
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\(\ 7 \times 3\ = 21\) |
\(\ 21 \div 3\ = 7\) |
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\(\ 7 \times 4\ = 28\) |
\(\ 28 \div 4\ = 7\) |
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\(\ 7 \times 5\ = 35\) |
\(\ 35 \div 5\ = 7\) |
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\(\ 7 \times 6\ = 42\) |
\(\ 42 \div 6\ = 7\) |
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\(\ 7 \times 7\ = 49\) |
\(\ 49 \div 7\ = 7\) |
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\(\ 7 \times 8\ = 56\) |
\(\ 56 \div 8\ = 7\) |
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\(\ 7 \times 9\ = 63\) |
\(\ 63 \div 9\ = 7\) |
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\(\ 7 \times 10\ = 70\) |
\(\ 70 \div 10\ = 7\) |
This allows students to establish the inverse relationship between multiplication and division.
Skill: Describing Patterns Among Whole Numbers up to 1000
Recognizing patterns is an important problem-solving skill; it facilitates the acquisition of other concepts and the formulation of conjectures leading to generalizations. The concept of patterns is the cornerstone of algebraic reasoning.
By observing and analyzing the relationships between numbers within a pattern, within a number sentence, or within the base-10 system, students discover patterns and can deepen their understanding of algebraic concepts.
Source: translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 16.
To get students to verbalize their observations, identify relationships, and explain how the use of patterning helped them in seeing the relationships between numbers, it is important that teachers ask relevant questions such as:
- What do you notice?
- What is repeated?
- What do you add? What are you taking away?
- What does this number represent?
- How could you represent this number differently?
- What is the link between your two representations?
- How have the number facts you know helped you calculate larger numbers?
- How do you know that multiplication and division are inverse operations? In what other situation might this help you?
To help students establish an intuitive understanding of the structure of the pattern, teachers encourage them to verbalize their observations.
In the example of the number 604, students can describe the inverse relationship of addition and subtraction by noticing, for example, that the numbers in the related operation are the same, but the order changes. Students can also describe their observations for addition that the sum always equals 604. As the first number increases by 1, the second number decreases by 1. Students can also describe their observations for subtraction by explaining that the difference increases by 1, the first number is always equal to 604, and the second number decreases by 1.
Using the example of the multiplication number facts and the associated division facts from the previous section, students can describe the pattern they noticed for the multiplication by noting that the product increases by 7, the first number remains constant, while the second number increases by 1. Students can describe the pattern they noticed for division by explaining that the first number increases by +7, the second number increases by 1, and the quotient remains constant at 7. The numbers are the same in the multiplication and associated division statements, but the order changes.