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 patterns on the basis of their constant rates and initial values
- comparing patterns:
- $1.00 deposited each day
- deposit doubled each day starting with $1.00
- deposit increased by $1.00 each day
- withdrawal of $1.00 each day starting with a balance of $10.00
- comparing linear growing patterns with different constant rates and the same initial value:
- starts with one tile at term 0 and grows by different amounts for each subsequent term:
- comparing linear growing patterns with the same constant rates and different initial values:
- starts with different amounts at term 0 and grows 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 the size of the elements 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.
- In a comparison of linear growing patterns, the pattern that has the greatest constant rate grows at a faster rate than the others and has a steeper incline as a line on a graph.
- The initial value (constant) of a linear growing 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 growing 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).
- 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.
Provide students with a variety of linear growing patterns that have the same initial value. Ask the students to compare the patterns, identifying how they are similar and how they are different:
- How are the lines the same?
- How are the lines different?
- What could these three lines represent?
Provide students with a variety of linear growing patterns that have different initial values but grow at the same rate. Ask the students to compare the patterns, identifying how they are similar and how they are different:
- 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 whole numbers and decimal numbers using various representations, including algebraic expressions and equations for linear growing patterns
- representations of a linear growing pattern involving whole numbers:
- representations of a linear growing pattern involving decimal numbers:
- representations of a shrinking pattern involving decimal numbers:
- Growing patterns are created by increasing the number of elements or the size of the elements in each iteration.
- A growing pattern can be created by repeating a pattern 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.
- 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 = 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.
Provide students with an algebraic equation for the nth term of a linear growing pattern that involves decimal numbers. Have them create a table of values for the first five terms of the pattern, then graph the line. Support them in making connections between the different representations by identifying the constant rate and the initial value in each representation.
Have students create and translate a variety of growing, shrinking, and non-linear patterns. It is important for students to stay flexible in their thinking about all kinds of patterns and how these can be represented in various ways.
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 involving whole numbers and decimal numbers, and use algebraic representations of the pattern rules to solve for unknown values in linear growing patterns
- For the tasks outlined below, students should begin by considering how many faces they can see in each term of the following pattern:
- extending patterns in multiple directions:
- What does term 0 look like? What does term 4 look like?
- making near and far predictions:
- How many faces are visible at term 0?
- How many faces are visible at term 10?
- How many faces are visible at term 100? 99? 101?
- identifying missing shapes, numbers in sequences and in tables of values, and points on a graph:
Term Number (n) |
Number of Visible Faces (F) |
0 | 3 |
1 | 5 |
2 | |
3 | 9 |
4 | |
5 | 13 |
43 | |
100 | |
n |
2n + 3 |
- using algebraic representations of pattern rules to solve for unknown values, such as F = 2n + 3, where F is the number of faces and n is the term number:
- How many faces are visible at term 9?
F = 2n + 3
= 2(9) + 3
= 18 + 3
= 21
- 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 interpolation.
- Determining a point beyond the graphical representation of a pattern is called extrapolation.
Ask students to extend the following table of values to include terms 0, 5, and 6:
Term Number | Term Value |
0 | |
1 | 10.00 |
2 | 5.00 |
3 | 2.50 |
4 | 1.25 |
5 | |
6 |
Have students make and test predictions about patterns so that they understand the role of pattern rules in making generalizations about a pattern, even when it is represented in various ways. For example, have students predict the distance travelled in 8 hours, using a strategy of their choice, given the following graph:
Provide students with different types of representations 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 the more possibilities for patterns will emerge. For example, have students identify the missing numbers in the sequence below:
3.412, ______, 5.414, ______, ______, 8.417, _____, 10.419
Provide students with different patterns represented in tables of values with missing elements, such as the one shown below. Ask students to determine:
- the value for a given position, such as term 20 in the table
- the term that has a given value, such as value 95 in the table
- the expression for the nth term, such as 3.2n + 5 in the table (shown for illustrative purposes only)
Term Number | Term Value |
0 | 5.0 |
1 | 8.2 |
2 | 11.4 |
3 | 14.6 |
4 | 17.8 |
20 | |
95.0 | |
n | 3.2n + 5 |
C1.4
create and describe patterns to illustrate relationships among integers
- number patterns to show the relationship between powers with base 2:
2 × 2 = 2^{2} = 4 |
2 × 2 × 2 = 2^{3} = 8 |
2 × 2 × 2 × 2 = 2^{4} = 16 |
2 × 2 × 2 × 2 × 2 = 2^{5} = 32 |
- number patterns to show the relationship between multiplication and division facts for 11:
11 × 1 = 11 | 11 ÷ 1 = 11 |
11 × 2 = 22 | 22 ÷ 2 = 11 |
11 × 3 = 33 | 33 ÷ 3 = 11 |
11 × 4 = 44 | 44 ÷ 4 = 11 |
11 × 5 = 55 | 55 ÷ 5 = 11 |
11 × 6 = 66 | 66 ÷ 6 = 11 |
11 × 7 = 77 | 77 ÷ 7 = 11 |
11 × 8 = 88 | 88 ÷ 8 = 11 |
11 × 9 = 99 | 99 ÷ 9 = 11 |
11 × 10 = 110 | 110 ÷ 10 = 11 |
11 × 11 = 121 | 121 ÷ 11 = 11 |
11 × 12 = 132 | 132 ÷ 12 = 11 |
- number patterns highlighting what happens to the integer signs when adding or subtracting integers:
2 + 4 = 6 | 5 − 4 = 1 |
2 + 3 = 5 | 5 − 3 = 2 |
2 + 2 = 4 | 5 − 2 = 3 |
2 + 1 = 3 | 5 − 1 = 4 |
2 + 0 = 2 | 5 − 0 = 5 |
2 + (−1) = 1 | 5 − (−1) = 6 |
2 + (−2) = 0 | 5 − (−2) = 7 |
2 + (−3) = −1 | 5 − (−3) = 8 |
2 + (−4) = −2 | 5 − (−4) = 9 |
- Patterns can be used to demonstrate relationships within and among number properties, such as expressing numbers in exponential 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 string pattern based on a key mathematical concept, such as the relationship between powers for different bases or the rules for addition and subtraction of integers. Ask students to complete the pattern. For example:
2 + 4 = 6 | 5 − 4 = 1 |
2 + 3 = 5 | 5 − 3 = 2 |
2 + 2 = 4 | 5 − 2 = 3 |
2 + 1 = 3 | 5 − 1 = 4 |
2 + 0 = 2 | 5 − 0 = 5 |
2 + (−1) = 1 | 5 − (−1) = 6 |
2 + (−2) = 0 | 5 − (−2) = 7 |
2 + (−3) = −1 | 5 − (−3) = 8 |
2 + (−4) = −2 | 5 − (−4) = 9 |
Have students use number patterns to show the multiplication and division facts for 11 and 12. For example:
11 × 1 = 11 | 11 ÷ 1 = 11 |
11 × 2 = 22 | 22 ÷ 2 = 11 |
11 × 3 = 33 | 33 ÷ 3 = 11 |
11 × 4 = 44 | 44 ÷ 4 = 11 |
11 × 5 = 55 | 55 ÷ 5 = 11 |
11 × 6 = 66 | 66 ÷ 6 = 11 |
11 × 7 = 77 | 77 ÷ 7 = 11 |
11 × 8 = 88 | 88 ÷ 8 = 11 |
11 × 9 = 99 | 99 ÷ 9 = 11 |
11 × 10 = 110 | 110 ÷ 10 = 11 |
11 × 11 = 121 | 121 ÷ 11 = 11 |
11 × 12 = 132 | 132 ÷ 12 = 11 |
Ask them to describe the patterns of the digits and explain how these patterns can help them recall these facts.