C1. Patterns and Relationships
Specific Expectations
Patterns
C1.1
identify and describe repeating, growing, and shrinking patterns, including patterns found in reallife contexts, and specify which growing patterns are linear
 linear growing pattern:
 Sample 1:
 starts with three tiles in position 1 (constant) and grows by four tiles in each subsequent position:
 Sample 1:
 Sample 2:
 starts with four tiles in position 1 (constant) and grows by three tiles in each subsequent position:
 nonlinear growing pattern:
 starts with three tiles in position 1, and the quantity of tiles added changes in each subsequent position:
 Repeating patterns have a pattern core that is repeated over and over.
 In growing patterns, there is an increase in the number of elements or the size of the elements from one term to the next.
 Some linear growing patterns have a direct relationship between the term number and the term value; for example, a pattern where each term value is four times its term number. Growing patterns that are linear can be plotted as a straight line on a graph.
 Each iteration of a 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. The term value is dependent on the term number. The relationship between the term number and the term value can be generalized.
 In shrinking patterns, there is a decrease in the number of elements or the size of the elements from one term to the next.
Note
 Growing and shrinking patterns are not limited to linear patterns.
 Many reallife objects and events can be viewed as having more than one type of pattern.
Have students collect pictures or make diagrams of patterns that they find in real life and create a pattern wall. Have them describe the regularities that they see in any of the patterns. Support students in recognizing that some reallife examples may be viewed in more than one way. For example, it is possible to think of depositing $1 in a bank every day as a repeating pattern. It is also possible to think of it as a growing pattern because the amount in the bank grows by $1 each day.
Provide students with different growing patterns represented in different ways. Ask them to identify whether the patterns are linear or nonlinear. For example:
Concrete Representation  Table of Values  Graph  





C1.2
create and translate repeating, growing, and shrinking patterns using various representations, including tables of values, graphs, and, for linear growing patterns, algebraic expressions and equations
 representations of a nonlinear growing pattern:
 representations of a linear shrinking pattern:
 representations of a linear growing pattern:
 Growing patterns are created by increasing the number of elements or the size of the elements in each iteration (term).
 A growing pattern can be created by repeating a pattern’s core. Each iteration shows how the total number of elements grows with each addition of the pattern core.
 Shrinking patterns are created by decreasing the number of elements or the size of the elements in each iteration.
 In translating a pattern from a concrete representation to a graph, 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. In a table of values, the term number is shown in the lefthand column and the term value is shown in the righthand column.
 A linear growing pattern can be represented using an algebraic expression or equation to show the relationship between the term number and the term value.
 Examining possible physical structures of a linear growing pattern can provide insight into the different algebraic equations that show the relationship between the term number and term value. For example, in Diagram 1, each term value can be viewed as four more than double the term number, which can be expressed as term value = 2*(term number) + 4 or y = 2x + 4.
Diagram 1
 Diagram 2 shows that for the same pattern, each term value can also be viewed as twice the term number plus two, which can be expressed as term value = term number + two + term number + two or y = x + 2 + x + 2. This expression for Diagram 2 can be simplified to y = 2x + 4, which is the same expression derived for Diagram 1.
Diagram 2
Note
 The creation of growing and shrinking patterns in this grade is not limited to linear patterns.
 The general equation for a linear growing pattern is y = mx + b, where x represents the term number, m represents the value of the multiplier, b represents a constant value, and y represents the term value.
Provide students with a table of values for a linear growing pattern in sequential order, such as the one provided below. Ask them to represent the pattern with colour tiles, highlighting what stays the same and what changes from one position to the next. Once students have completed their concrete representation of the pattern, ask them to represent the same pattern on a graph.
Position Number  Number of Tiles 
0  5 
1  8 
2  11 
3  14 
4  17 
5  20 
6  23 
7 
26 
In the consolidation of the activity, support students in recognizing that:
 the constant of 5 is first seen at position 0 and stays the same throughout the pattern;
 the change in the pattern is that it increases by 3 in each position.
Ask students questions such as:
 What position uses 50 tiles?
 How many tiles would you need to represent the 100th position?
 What is the connection between the position number and the number of tiles?
 What will happen to the pattern if position 0 has a value of 3?
 What will happen to the pattern if groups of 5 are added each time?
 How does changing the “groups of” (the multiplier) in the pattern change the pattern rule?
Ask students to represent the new patterns graphically and to look for connections between the different representations and patterns (e.g., What is the same between them? What is different between them?).
Students should have many opportunities to create and represent linear growing patterns and to test out their conjectures for what will happen in new situations. Eventually, after many opportunities, support students in naming features of linear patterns as “the constant” and “the multiplier”.
Provide students with a table of values for a linear growing pattern that is not ordered and has only some of the values shown, such as the table below. Have them determine the relationship between the position number and the position value and express this relationship as an algebraic expression that can be used to determine any position value. With this task, students use functional thinking.
Position Number  Position Value 
1  9 
6  
30  
37  
2  16 
3  23 
7  51 
n 
C1.3
determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating, growing, and shrinking patterns, and use algebraic representations of the pattern rules to solve for unknown values in linear growing patterns
 extending patterns in multiple directions:
 What does position 0 look like? What does position 5 look like?
 making near and far predictions:
 How many tiles are needed to show position 0?
 How many tiles are needed to show position 10?
 How many tiles are needed to show position 100? 99? 101?
 identifying missing shapes, numbers in number sequences and tables of values, and points on graphs:
 using algebraic representations of pattern rules to solve for unknown values, such as T = 3p, where T is the number of tiles and p is the position number:
 How many tiles are in position 10?
T = 3p
= 3 × 10
= 30
 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, showing what comes next and what came before.
 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 that 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 that can be justified.
 Identifying the missing elements in a pattern represented using a table of values may require determining the term number (x) or the term value (y).
 Identifying the missing elements in a pattern represented on a graph may require determining the point (x, y) within the given representation or beyond it, in which case the pattern will need to be extended.
 The algebraic expression that represents a linear growing pattern is also referred to as the general term or the nth term. It can be used to solve for the term value or the term number.
Note
 Determining a point within the graphical representation of a pattern is called interpolating.
 Determining a point beyond the graphical representation of a pattern is called extrapolating.
Ask students to extend patterns represented in different ways. For example:
 Extend the graph to show the number of tiles for position 5:
 Extend the table to show the number of tiles for positions 5, 6, 10, and n:
Position Number  Number of Tiles 
1  2 
2  5 
3  8 
4  11 
.  . 
.  . 
.  . 
. 
Have students make and test predictions about patterns so that they understand the role of pattern rules in making generalizations about patterns represented in various ways. For example, ask them to predict how much money will be in the bank on the 13th day, if the pattern below is followed, and then verify their prediction. They can interpret the dollar amounts as either deposits, or the value of the money in the bank on that day.
Provide students with different patterns represented in tables of values with missing elements, such as the one below, and ask them to:
 determine missing values; for example, what is the value for position 0? 20?
 determine missing positions; for example, which position has a value of 11? 95?
Position Number  Position Value 
0  
1  8 
11  
3  14 
4  17 
20  
95 
C1.4
create and describe patterns to illustrate relationships among whole numbers and decimal numbers
 number patterns to show the relationships between ones, tenths, hundredths, and thousandths:
Sample 1:
3.271 =  + 3 ones  + 2 tenths  + 7 hundredths  + 1 thousandth 
3.271 =  + 3 ones  + 2 tenths  + 6 hundredths  + 11 thousandths 
3.271 =  + 3 ones  + 2 tenths  + 5 hundredths  + 21 thousandths 
3.271 =  + 3 ones  + 2 tenths  + 4 hundredths  + 31 thousandths 
3.271 =  + 3 ones  + 2 tenths  + 3 hundredths  + 41 thousandths 
3.271 =  + 3 ones  + 2 tenths  + 2 hundredths  + 51 thousandths 
3.271 =  + 3 ones  + 2 tenths  + 1 hundredth  + 61 thousandths 
3.271 =  + 3 ones  + 2 tenths  + 0 hundredths  + 71 thousandths 
Sample 2:
3.271 =  + 3 ones  + 2 tenths  + 7 hundredths  + 1 thousandth 
3.271 =  + 3 ones  + 1 tenth  + 17 hundredths  + 1 thousandth 
3.271 =  + 3 ones  + 0 tenths  + 27 hundredths  + 1 thousandth 
.  .  .  .  . 
.  .  .  .  . 
.  .  .  . 
 number patterns to show the relationships between addition and subtraction facts for 7 when applied to decimal thousandths:
5.000 + 0.007 = 5.007  5.007 − 0.007 = 5.000 
5.001 + 0.006 = 5.007  5.007 − 0.006 = 5.001 
5.002 + 0.005 = 5.007  5.007 − 0.005 = 5.002 
5.003 + 0.004 = 5.007  5.007 − 0.004 = 5.003 
5.004 + 0.003 = 5.007  5.007 − 0.003 = 5.004 
5.005 + 0.002 = 5.007  5.007 − 0.002 = 5.005 
5.006 + 0.001 = 5.007  5.007 − 0.001 = 5.006 
5.007 + 0.000 = 5.007  5.007 − 0.000 = 5.007 
 Patterns can be used to demonstrate relationships among numbers.
 There are many patterns within the decimal number system.
Note
 Many number strings are based on patterns and on the use of patterns to develop a mathematical concept.
 The use of the word “strings” in coding is different from its use in “number strings”.
Provide students with a partial number pattern based on a key mathematical concept, such as understanding place value. Have them continue the pattern to rename a number, such as 3.271, in as many ways as they can in terms of ones, tenths, hundredths, and thousandths. Students may notice that when they take away 1 hundredth, they have to put back 10 thousandths.
Ask students to create a fourdigit number with three decimal places. Have them cover their number using counters on a Gattegno chart. (A sample Gattegno chart can be downloaded at BLM: Grade 6 C1.4 Gattegno chart.) Then ask them to:
 multiply their number by 100 and move their counters so that they represent the new number (each counter goes up two rows, but stays in the same column);
 predict what will happen if they multiply their new number by 100;
 verify their prediction and move their counters to the new position;
 determine how they can get their counters back to the original positions using division.
Next, ask students to “switch gears” and think about how they can move the counters that represent their original number up two rows using division instead of multiplication.
Continue having students explore the patterns of multiplying (and dividing) by 10, 100, 1000, and 10 000 and by 0.1, 0.01, and 0.001, and support them in making generalizations about the results.