## C1. Patterns and Relationships

### Specific Expectations

#### Patterns

C1.1

identify and compare a variety of repeating, growing, and shrinking patterns, including patterns found in real-life contexts, and compare linear growing and shrinking patterns on the basis of their constant rates and initial values

**comparing linear growing and shrinking patterns with the same initial value:**- starts with four tiles at term 0 and changes by different amounts for each subsequent term:

**comparing linear shrinking patterns with the same constant rate:**- starts with different amounts at term 0 and shrinks by one tile for each subsequent term:

- 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 size from one term to the next.
- In shrinking patterns, there is a decrease in the number of elements or size from one term to the next.
- If the ratio of the change in one variable to the change in another variable is equivalent between any two sets of data points, then there is a constant rate. An example of a real-life application of a constant rate is an hourly wage of $15.00 per hour.
- The initial value (constant) of a linear pattern is the value of the term when the term number is zero. An example of a real-life application of an initial value is a membership fee.
- The relationship between the term number and the term value can be generalized. A linear pattern of the form
*y*=*mx*+*b*has a constant rate,*m,*and an initial value,*b*. The graph of a linear growing pattern that has an initial value of zero passes through the origin at (0, 0). - The graphical representation of a linear
*growing*pattern is a line that rises to the right; for a linear*shrinking*pattern, the line descends to the right.

*Note*

- Growing and shrinking patterns are not limited to linear patterns.

Provide students with real-life patterns that can be compared, such as heart rate variability (HRV), shown below:

Ask students to describe how the patterns are the same and how they are different. Discuss how analysing these patterns can provide insight and be used to make predictions.

Provide students with a variety of patterns, including some with the same initial values:

Ask:

- How are the lines the same?
- How are the lines different?
- What could these three lines represent?

Provide students with a variety of linear shrinking patterns that have different initial values but shrink at the same rate.

Ask:

- How are the lines the same?
- How are the lines different?
- What could these three lines represent?

C1.2

create and translate repeating, growing, and shrinking patterns involving rational numbers using various representations, including algebraic expressions and equations for linear growing and shrinking patterns

**representations of a linear shrinking pattern involving integers:**

**representations of a linear growing pattern involving integers:**

**representations of a shrinking pattern involving fractions:**

**representations of a linear growing pattern involving decimals:**

- Growing patterns are created by increasing the number of elements or the size of the elements in each iteration.
- Shrinking patterns are created by decreasing the number of elements or the size of the elements in each iteration.
- Graphical representations of linear growing and shrinking patterns appear as straight lines.
- Graphical representations of non-linear growing and shrinking patterns appear as curves.
- Some patterns are based on continuous variables, such as height, distance, or time. Graphical representations of continuous values are solid lines or curves, illustrating their continuous nature.
- A linear growing pattern can be created by repeatedly representing a pattern to show the total number of elements in each iteration of the pattern core.

- Examining the physical structure of a linear growing pattern can provide insight into the different algebraic equations that show the relationship between the term number and the 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*= 2*x*+ 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*= 2*x*+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.

Provide students with an algebraic equation, such as *y *= 1.5*x *+ 1, for a linear growing or shrinking pattern. Have them graph the line for the first five terms:

Have students create and translate a variety of growing and shrinking non-linear patterns. It is important for students to be flexible in their thinking about all kinds of patterns and about how patterns can be represented in various ways, such as with concrete materials, graphs drawn by hand or with technology, and tables of values

C1.3

determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in growing and shrinking patterns involving rational numbers, and use algebraic representations of the pattern rules to solve for unknown values in linear growing and shrinking patterns

**extending patterns in multiple directions:**- What does term −1 look like? What does term 5 look like?

**making near and far predictions:**- What is the value of the tiles for term 5? 10?
- What is the value of the tiles for term −5? −10?

**identifying missing elements in pictorial representations, geometric patterns, number sequences, tables of values, and graphs:**

**using algebraic representations of pattern rules to solve for unknown values, such as****V****= −2****n****− 3 where****V****is the term value and****n****is the term number:**- What is the term value at term 12?

*V *= –2*n* – 3

* *= –2(12) – 3

* *= –24 – 3

* *= −27

- 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 and shrinking pattern is also referred to as the general term or the
*n*th term. It can be used to solve for the term value or the term number.

*Note*

- Determining a point on a graph within a given set of points that fit a pattern is called interpolation. Determining a point on a graph beyond a given set of points that fit a pattern is called extrapolation. This skill set is used in a variety of contexts, including in the Grade 8 Structures and Mechanisms strand of the Science and Technology curriculum, 2022 (e.g., when students are working on the concept of the mechanical advantage and levers).

Ask students to extend the graph to include terms −1, 5, and 6:

Support students in recognizing the relationship between two sets of data, such that each element in a set is associated with an element in the other set. For example, in the equation 2*x *+ 3 = *y*, the *y*-values are calculated by replacing the *x* in the equation *y *= 2*x *+ 3 with different *x* values, as demonstrated in the following table of values:

x |
y |

−4 | −5 |

0 | 3 |

2 | 7 |

13 | |

9 | 21 |

Have students reflect on the following questions:

- What happens to
*y*when*x*increases by 1? How do you know? - What is the value of
*x*when*y*is 13? How do you know?

Provide students with different patterns represented in tables of values with missing elements, such as the table shown below. Ask them to determine:

- the value for a given term, such as term 20 in the table
- the term that has a given value, such as value −53 in the table
- the expression for the
*n*th term*,*such as −2*n*− 3 in the table

Term Number |
Term Value |

0 | −3 |

1 | −5 |

2 | −7 |

3 | −9 |

4 | −11 |

20 | |

−43 | |

n |
− 2n −3 |

C1.4

create and describe patterns to illustrate relationships among rational numbers

**number patterns to show the relationship between standard and scientific notation:**

2 × 10^{4} = 20 000 |

2 × 10^{3} = 2000 |

2 × 10^{2} = 200 |

2 × 10^{1} = 20 |

2 × 10^{0} = 2 |

2 × 10^{-1} = 0.2 |

2 × 10^{-2} = 0.02 |

**number patterns to highlight what happens to the signs when integers are multiplied:**

2 × 4 = 8 | −2 × 4 = −8 | 2 × (−4) = −8 | −2 × (−4) = 8 |

2 × 3 = 6 | −2 × 3 = −6 | 2 × (−3) = −6 | −2 × (−3) = 6 |

2 × 2 = 4 | −2 × 2 = −4 | 2 × (−2) = −4 | −2 × (−2) = 4 |

2 × 1 = 2 | −2 × 1 = −2 | 2 × (−1) = −2 | −2 × (−1) = 2 |

2 × 0 = 0 | −2 × 0 = 0 | 2 × 0 = 0 | −2 × 0 = 0 |

2 × (−1) = −2 | −2 × (−1) = 2 | 2 × 1 = 2 | (−2) × 1 = −2 |

2 × (−2) = −4 | −2 × (−2) = 4 | 2 × 2 = 4 | (−2) × 2 = −4 |

2 × (−3) = −6 | −2 × (−3) = 6 | 2 × 3 = 6 | (−2) × 3 = −6 |

2 × (−4) = −8 | −2 × (−4) = 8 | 2 × 4 = 8 | (−2) × 4 = −8 |

- Patterns can be used to demonstrate an understanding of number properties, including the use of exponents to express numbers in scientific notation.

*Note*

- Using patterns is a useful strategy in developing understanding of mathematical concepts, such as knowing what sign to use when two integers are added or subtracted.

Provide students with a partial number pattern based on a key mathematical concept such as the meaning of the exponents when numbers are expressed in scientific notation:

2 × 10^{4} = 20 000 |

2 × 10^{3} = 2000 |

2 × 10^{2} = 200 |

2 × 10^{1} = 20 |

2 × 10^{0} = 2 |

2 × 10^{-1} = 0.2 |

2 × 10^{-2} = 0.02 |

Have students use number patterns to show the multiplication and division of integers. Ask them to describe the patterns of the signs, and ask how they can apply these patterns to cases where more than two integers are multiplied together:

2 × 4 = 8 | −2 × 4 = −8 |

2 × 3 = 6 | −2 × 3 = −6 |

2 × 2 = 4 | −2 × 2 = −4 |

2 × 1 = 2 | −2 × 1 = −2 |

2 × 0 = 0 | −2 × 0 = 0 |

2 × (−1) = −2 | −2 × (−1) = 2 |

2 × (−2) = −4 | −2 × (−2) = 4 |

2 × (−3) = −6 | −2 × (−3) = 6 |

2 × (−4) = −8 | −2 × (−4) = 8 |