C1. Patterns and Relationships
Specific Expectations
Patterns
C1.1
identify and describe repeating elements and operations in a variety of patterns, including patterns found in real-life contexts
- patterns with repeating elements or repeating operations:
- handicrafts: quillwork, beadwork, knitting, crocheting, quilt patches, embroidery, visual motifs
- nature: honeycombs, spiderwebs, phases of the Moon:
- 25 cents put in a piggy bank every sixth day
- Patterns may involve a repeating element and a repeating operation (e.g., in a design, different sizes of squares may be repeated).
- The shortest string of elements that repeat in a pattern is referred to as the “pattern core”.
- The quantifying measure or numerical value in a pattern may involve a repeat of addition, subtraction, multiplication, or division.
Note
- Students can engage in mathematics and patterns through the contexts, cultural histories, and stories of various cultures.
- Have students focus on how attributes are staying the same and how they are changing.
- A repeat operation involving addition and subtraction of zero will result in a pattern whose elements are not altered.
- A repeat operation involving multiplication and division by one will result in a pattern whose elements are not altered.
Create a class pattern wall. Have students collect pictures or make diagrams of repeating patterns that they find in real life. For each pattern, have students write descriptions of which elements repeat and how. For example, students may find patterns in beadwork:
Give students opportunities to explore written number systems in different cultures. This will enhance their understanding that different symbols can be used to represent numbers. For example, have students look for patterns in the Mayan number system, illustrated below, and discuss how the patterns connect to the numbers (e.g., the 9 is represented by a combination of the 4 and 5 symbols):
Similar patterns can be found in other written number systems from other ancient cultures.
C1.2
create and translate patterns that have repeating elements, movements, or operations using various representations, including shapes, numbers, and tables of values
- translating a numeric pattern with repeating elements into a geometric pattern using shapes:
- creating a pattern with a repeating operation using tiles and translating it into a table of values:
- The same pattern structure can be represented in various ways.
- Patterns with a repeating element can be based on attributes (e.g., colour, size, orientation).
- Patterns with a repeating operation can be based on repeating operations of addition, subtraction, multiplication, and/or division.
- Pattern structures can be generalized.
- When translating a pattern from a concrete representation to a table of values, each iteration of the pattern can be referred to as the term number, and the number of elements in each iteration can be referred to as the term value. In a table of values, the term number is shown in the left-hand column and the term value is shown in the right-hand column.
Note
- Comparing translated patterns highlights the equivalence of their underlying mathematical structure, even though the representation differs.
Have students describe patterns that are relevant to their lives, including those with historical and cultural significance; for example, “Every year on my birthday, I get one year older.”
Give students a repeating numeric pattern and ask them to translate it into a geometric pattern using shapes.
Have students create a growing pattern where the pattern elements grow with each iteration. For example, in the pattern below, the yellow square and the red square repeat and increase by one for each term.
Have students translate the pattern into a table of values, identifying the number of tiles in each iteration of the pattern. For example:
Ask students follow-up questions such as:
- How many tiles would be at the 6th position?
- If I had 16 tiles, what position number could I show?
C1.3
determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in patterns that have repeating elements, movements, or operations
- extending patterns in multiple directions:
- What came before?
- What comes next?
- making near and far predictions:
- What shape would be in the 10th position?
- What shape would be in the 20th position?
- What shape would be in the 100th position? The 99th position? The 101st position?
- identifying missing shapes, numbers in sequences, and numbers in tables of values for patterns:
Number of Full Ten Frames |
Number of Counters |
1 |
10 |
2 |
20 |
3 |
|
4 |
|
5 |
|
|
60 |
|
70 |
- Patterns can be extended because they are repetitive by nature.
- Pattern rules are generalizations about a pattern, and they can be described in words.
- Patterns can be extended in multiple directions – up, down, right, left, diagonally.
- To make a near prediction about a pattern is to state or show what a pattern will look like just beyond the given representation of that pattern. The prediction can be verified by extending the pattern.
- To make a far prediction about a pattern is to state or show what a pattern will look like well beyond the given representation of that pattern. Often calculations are needed to make an informed prediction or to verify a prediction.
- To identify missing elements of patterns is to complete a representation for a given pattern by filling in the missing parts.
Note
- In order to extend, predict, or determine missing elements, students need to generalize patterns, using pattern rules.
- Rules should be used to verify predictions and to critically analyse extensions and solutions for missing elements.
Give students patterns that do not start at position 1, like the one in the diagram below. Have them extend the patterns in both directions, in this case showing positions 1, 2, 13, and 14.
Give students patterns that are arranged in multiple directions, such as the one shown below, and ask them to think about the patterns in more than one way. Have them extend the patterns for the next row of numbers and justify their choice of numbers.
Have students make and test predictions about various patterns so they understand the role of pattern rules in making generalizations. Provide them with a variety of patterns, and have them make near and far predictions. For example, for the pattern below, ask them how many circle sections will appear in the 100th position, and have them explain the rule they used to arrive at the answer.
Also ask questions like: “If I had 50 circle sections, could I show a position number in the pattern? Why or why not?”
Have students extend a variety of patterns. For example, for the pattern below, ask them to show positions 1, 2, 13, and 14. Ask them questions such as: “Which elements will be in the 100th position? The 101st position? The 99th position?” Have them justify their predictions.
Provide students with different types of patterns with missing elements so that they can think critically about possible pattern rules based on the information they are given. The more information that is missing, the more thinking and possibilities for patterns will emerge. For example:
- Use additional green, blue, orange, and red squares to extend the pattern. You can use more than one of each colour.
- What numbers are missing in this pattern?
C1.4
create and describe patterns to illustrate relationships among whole numbers up to 1000
- creating patterns to show the relationship between addition and subtraction facts for 7 in larger numbers:
500 + 7 = 507 |
507 − 7 = 500 |
501 + 6 = 507 |
507 − 6 = 501 |
502 + 5 = 507 |
507 − 5 = 502 |
503 + 4 = 507 |
507 − 4 = 503 |
504 + 3 = 507 |
507 − 3 = 504 |
505 + 2 = 507 |
507 − 2 = 505 |
506 + 1 = 507 |
507 − 1 = 506 |
507 + 0 = 507 |
507 − 0 = 507 |
- creating patterns to show the relationship between multiplication and division facts for 5:
5 × 1 = 5 |
5 ÷ 1 = 5 |
5 × 2 = 10 |
10 ÷ 2 = 5 |
5 × 3 = 15 |
15 ÷ 3 = 5 |
5 × 4 = 20 |
20 ÷ 4 = 5 |
5 × 5 = 25 |
25 ÷ 5 = 5 |
5 × 6 = 30 |
30 ÷ 6 = 5 |
5 × 7 = 35 |
35 ÷ 7 = 5 |
5 × 8 = 40 |
40 ÷ 8 = 5 |
5 × 9 = 45 |
45 ÷ 9 = 5 |
5 × 10 = 50 |
50 ÷ 10 = 5 |
- Patterns can be used to understand relationships among numbers.
- There are many patterns within the whole number system.
Note
- Many number strings are based on patterns and the use of patterns to develop a mathematical concept.
Have students create their own number string patterns to illustrate a relationship among numbers. Ask them to justify their thinking. For example, they may demonstrate how the addition facts for 7 can be used to add large numbers, or they may demonstrate the patterns for multiplication and division by 5.
500 + 7 = 507 |
507 − 7 = 500 |
501 + 6 = 507 |
507 − 6 = 501 |
502 + 5 = 507 |
507 − 5 = 502 |
503 + 4 = 507 |
507 − 4 = 503 |
504 + 3 = 507 |
507 − 3 = 504 |
505 + 2 = 507 |
507 − 2 = 505 |
506 + 1 = 507 |
507 − 1 = 506 |
507 + 0 = 507 |
507 − 0 = 507 |
5 × 1 = 5 |
5 ÷ 1 = 5 |
5 × 2 = 10 |
10 ÷ 2 = 5 |
5 × 3 = 15 |
15 ÷ 3 = 5 |
5 × 4 = 20 |
20 ÷ 4 = 5 |
5 × 5 = 25 |
25 ÷ 5 = 5 |
5 × 6 = 30 |
30 ÷ 6 = 5 |
5 × 7 = 35 |
35 ÷ 7 = 5 |
5 × 8 = 40 |
40 ÷ 8 = 5 |
5 × 9 = 45 |
45 ÷ 9 = 5 |
5 × 10 = 50 |
50 ÷ 10 = 5 |