C1. Patterns and Relationships
Specific Expectations
Patterns
C1.1
identify and describe repeating and growing patterns, including patterns found in reallife contexts
 repeating patterns in real life:
Snowflake  Tiles 
Windows in a Skyscraper  Brick Wall 
Border  Fence 

 growing patterns in real life:
Shell  Wings 


Leaf  Mayan Temple 

 The complexity of a repeating pattern depends on:
 the nature of the attribute(s);
 the number of changing attributes;
 the number of elements in the core of the pattern;
 the number of changing elements within the core.
 In growing patterns, there is an increase in the number of elements or the size of the elements from one term to the next.
Note
 Students can engage in mathematics and patterns through the contexts, cultural histories, and stories of various cultures.
Start a pattern wall. Have students collect pictures or make diagrams of patterns that they find in real life to add to the wall. Have them describe the regularities that they see in the patterns. Some reallife examples may be viewed in more than one way; some students may see the elements that repeat and others might see how the pattern is growing. If geometric patterns that have cultural significance are shared, use the opportunity to discuss the history and meaning of the patterns.
C1.2
create and translate repeating and growing patterns using various representations, including tables of values and graphs
 creating a repeating pattern using tiles or beads, then translating it into a numeric representation of each row in the pattern:
 creating a growing pattern using concrete materials, then translating it into a table of values:
 translating a geometric growing pattern into a table of values and a graph:
 The same pattern structure can be represented in various ways.
 Repeating patterns can vary in complexity, but all are created by iterating their pattern core.
 Growing patterns are created by increasing the number of elements in each iteration.
 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 lefthand column and the term value is shown in the righthand column.
 The term value is dependent on the term number. The term number (x) is represented on the horizontal axis of the Cartesian plane, and the term value (y) is represented on the vertical axis. Each point (x, y) on the Cartesian plane is plotted to represent the pattern.
Note
 The creation of growing patterns in this grade is not limited to linear patterns.
 For (x, y), the xvalue is the independent variable and the yvalue is the dependent variable.
 Comparing translated patterns highlights the equivalence of their underlying mathematical structure, even though the representations differ.
Repeating patterns involving shapes can vary in complexity, yet they are all created by iterating a pattern core. Have students create pattern cores that range from a simple repeating pattern involving one attribute to more complex ones involving multiple attributes. Then have them iterate their pattern cores at least three times. Have them share with others in the class and identify the changing attributes in each other’s work.
Have students make a growing pattern with square tiles. Then have them translate their pattern to grid paper. Have them place a dot at the top of each bar (see illustration below) to make connections to learning about graphs. Ask students to write about the connections that they can make between the two representations.
Growing patterns can grow in various ways. The numbers in a growing pattern increase in size from one term to the next, but the way they increase can vary. Have students create a growing pattern and translate it into a number sequence, a table of values, and a graph. Support them in making connections between the table of values, the graph, and the way the numbers are growing.
C1.3
determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating and growing patterns
 extending patterns in multiple directions:
 What comes next? What came before?
 What comes next? What came before?
 making near and far predictions:
 How many blocks are in the 5th position? The 10th position? The 100th position?
 identifying missing shapes, numbers in a number sequence and in a table of values, and points on a graph:
 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.
Ask students to extend patterns given any type of representation. For example, have them describe the pattern rule to extend the pattern to the next three terms in the pictorial representation below.
Ask followup questions such as:
 If I had exactly 36 tiles, what position number could I show?
 Can I show a position number with 55 tiles? Why or why not?
Have students determine pattern rules to make and justify predictions about patterns represented in various ways. For example, present a fictional scenario about a teacher and a class.
 A teacher asked their students to state the pattern rule for the following pattern:
Here are two responses:
Mishaal said, “I see two rows of circles. In the top row, there is always one more circle (in blue) than the position number. In the bottom row, there are always two more circles (in grey) than the position number.” 
Jennifer said, “To find the total number of circles, you need to double the position number (yellow circles) and add three (blue circles). So, for position 1, it is 1 + 1 + 3 = 5; for position 2, it is 2 + 2 + 3 = 7; and for position 3, it is 3 + 3 + 3 = 9.” 
Ask: Which rule is correct? Can you find another rule that works for the pattern?
Have students continue the pattern for each of the scenarios outlined below.
 extending a pattern presented in a graph:
 extending a pattern presented with concrete materials:
 extending a pattern in a table of values:
Provide students with different types of patterns that have 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, ask students to colour in the squares to complete the pattern below, then describe how the pattern could be extended in at least two directions:
Provide a variety of number patterns that enable students to practise extending and finding missing terms. For example:
 number sequences where the same addend is used:
 3, 6, 9, 12, ____, ____, ____, …
 number sequences where the same multiplier is used:
 1, 2, 4, ____, ____, 32, ____, …
 number sequences where the addend increases from term to term:
 1, 2, 4, 7, 11, ____, ____, 29, ____, ...
 special number sequences:
 0, 1, 4, 9, ____, ____, ____, ... (square number sequence)
 1, 1, 2, 3, 5, ____, ____, ____, 34, 55, 89 (Fibonacci sequence)
C1.4
create and describe patterns to illustrate relationships among whole numbers and decimal tenths
 creating a pattern to show the relationship between the place values of digits:
 when the tens place goes down by 1, the ones place goes up by 10:
37.1 = 
3 tens 
+ 7 ones 
+ 1 tenth 
37.1 = 
2 tens 
+ 17 ones 
+ 1 tenth 
37.1 = 
1 ten 
+ 27 ones 
+ 1 tenth 
37.1 = 
0 tens 
+ 37 ones 
+ 1 tenth 
 when the ones place goes down by 1, the tenths place goes up by 10:
37.1 = 
3 tens 
+ 7 ones 
+ 1 tenth 
37.1 = 
3 tens 
+ 6 ones 
+ 11 tenths 
37.1 = 
3 tens 
+ 5 ones 
+ 31 tenths 
37.1 = 
3 tens 
+ 4 ones 
+ 31 tenths 
. . . 
. . . 
. . . 
. . . 
 creating patterns to show the relationship between addition and subtraction facts for 7 tenths:
5.0 + 0.7 = 5.7 
5.7 − 0.7 = 5.0 
5.1 + 0.6 = 5.7 
5.7 − 0.6 = 5.1 
5.2 + 0.5 = 5.7 
5.7 − 0.5 = 5.2 
5.3 + 0.4 = 5.7 
5.7 − 0.4 = 5.3 
5.4 + 0.3 = 5.7 
5.7 − 0.3 = 5.4 
5.5 + 0.2 = 5.7 
5.7 − 0.2 = 5.5 
5.6 + 0.1 = 5.7 
5.7 − 0.1 = 5.6 
5.7 + 0.0 = 5.7 
5.7 − 0.0 = 5.7 
 creating patterns to show the relationship between multiplication and division facts:
9 × 1 = 9 
9 ÷ 1 = 9 
9 × 2 = 18 
18 ÷ 2 = 9 
9 × 3 = 27 
27 ÷ 3 = 9 
9 × 4 = 36 
36 ÷ 4 = 9 
9 × 5 = 45 
45 ÷ 5 = 9 
9 × 6 = 54 
54 ÷ 6 = 9 
9 × 7 = 63 
63 ÷ 7 = 9 
9 × 8 = 72 
72 ÷ 8 = 9 
9 × 9 = 81 
81 ÷ 9 = 9 
9 × 10 = 90 
90 ÷ 10 = 9 
 Patterns can be used to understand relationships between whole numbers and decimal numbers.
Note
 Many number strings are based on patterns and the use of patterns to develop a mathematical concept.
Provide students with a partial number string pattern involving place value. Have them continue the string to rename a number such as 37.1 in as many ways they can using different tens, ones, and tenths. Support students in noticing that for each decrease of 10 ones, there is an increase of 10 tenths.
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 and subtraction facts for 7 can be used when working with decimal tenths, or they could demonstrate the pattern for multiplication and division by 9.