C1. Patterns and Relationships:
identify, describe, extend, create, and make predictions about a variety of patterns, including those found in real-life contexts
identify and describe repeating, growing, and shrinking patterns, including patterns found in real-life contexts, and specify which growing patterns are linear
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 real-life 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.
Coin image © 2022 Royal Canadian Mint. All rights reserved.
Provide students with different growing patterns represented in different ways. Ask them to identify whether the patterns are linear or non-linear. For example:
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
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.
In the consolidation of the activity, support students in recognizing that:
Ask students questions such as:
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.
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
T = 3p
= 3 × 10
Ask students to extend patterns represented in different ways. For example:
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:
create and describe patterns to illustrate relationships among whole numbers and decimal numbers
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 four-digit 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:
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.