Patterns and Algebra
Across K-7, students are developing big ideas that connect patterns and algebra to other areas of mathematics such as number and operations. Students learn to identify regularities whether in repeating patterns or changes in increasing or decreasing patterns and generalize what is happening mathematically such as being able to predict what comes next. Students learn to look for number relationships when exploring a variety of patterns, including numbers in a hundreds chart, visual patterns, and patterns in art, music and nature. Students develop algebraic thinking across the grades by making generalizations, looking for or creating patterns and seeking number relationships and learn to notate these relationships using symbols that include expressions and graphing. Other big concepts that develop across K-7 include the meanings of equality and inequality, change, and solving for unknowns.
As students explore patterns and mathematical relationships there are many opportunities to connect to students’ lives, community, culture, and place. With these experiences we are honouring the following First Peoples Principle of Learning: Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).
As we learn about key concepts in patterns and algebra, we will also be developing many curricular competencies. Three that we have chosen to focus on in our designing of lesson ideas are:
- Represent mathematical ideas in concrete, pictorial and symbolic forms
- Connect mathematical concepts to each other, other areas of learning and personal interests
- Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving
Although these three curricular competencies have been highlighted, there will be many opportunities to develop many curricular competencies during the investigation of patterns and algebra.
Learning Story for Grade 6
Patterns and Algebra
In prior grades, students think about “What comes next?” recursively. For example, they might describe 5, 8, 11, … using the pattern rule “start at five, add three each time”; in order to determine the 7th number in the sequence, they might extend the pattern to 5, 8, 11, 14, 17, 20, 23. In Grade 6, students are encouraged to think about increasing and decreasing patterns as functional relationships. That is, they begin to relate inputs and outputs. For example, they might describe 5, 8, 11, … using the expression 3n + 2; in order to determine the 100th number in the sequence, they might add two to one-hundred groups of three. In Grade 6, patterns are limited to those that involve whole numbers. In future grades, students will extend this thinking to patterns that involve integers and rational numbers.
Grade 6 also marks the first time that students represent increasing and decreasing patterns using expressions and graphs. (The learning story for Grade 6 Data and Probability introduces students to line graphs.) Students learn that there are many different ways to represent the same pattern and that each way has its purpose. Because patterns are limited to those that involve whole numbers, graphs are limited to the first quadrant.
In Grade 6, students build on their understanding of one-step equations from previous grades. In Grade 3, students solve one-step equations that involve additions or subtraction; in Grade 4, students solve one-step equations that involve all four operations. In Grade 5, students represent unknown quantities using variables, rather than blank spaces or boxes. Methods involve using number sense and mental mathematics. In Grade 6, students use inverse operations (e.g., subtraction “undoes” addition) and the preservation of equality (i.e., subtracting three from both sides of an equation maintains “balance”).
Increasing and decreasing patterns, using expressions, tables, and graphs as functional relationships
Students begin to make predictions using functional–rather than recursive–relationships. They are introduced to expressions, tables, and graphs as different ways of representing the same relationship. Students will explore increasing and decreasing patterns that involve concrete materials, contextualized situations, and number sequences.
One-step equations with whole-number coefficients and solutions
Students continue to solve one-step equations that involve whole numbers. They are introduced to inverse operations and the preservation of equality.
Key Number Concept 1: increasing and decreasing patterns, using expressions, tables, and graphs as functional relationships
The shift from thinking about patterns as recursive to functional relationships is key to students developing an understanding of functions (and proportional reasoning) in later grades. When students look for a relationship that leads to an output for any given input, they are developing algebraic reasoning. This process of thinking across a table of values is seen as more efficient than extending a table of values one row at a time, particularly when answering “What comes ‘way down the line’?” Representing increasing and decreasing patterns using graphs highlights relationships visually. Students see that these increasing and decreasing patterns–or linear relationships–always result in a line when graphed. Hence, linear relationships can be recognized at a glance. Other representations have their purposes: tables might be useful to determine the increase or decrease; expressions might be useful to determine a particular output for a particular input.
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of increasing and decreasing patterns in Grade 6:
- increasing and decreasing patterns in tables and charts
- rules (words) for increasing and decreasing patterns
- represent increasing and decreasing patterns visually and numerically
- explore patterns that involve multiplicative rules (e.g., “the number of tiles is equal to the figure number times two”)
- explore patterns that involve composite rules (e.g., “the number of tiles is five times the figure number and one more”)
- develop expressions to represent pattern rules
- show increasing patterns as functions that relate input and output
- generalize relationships from contextualized situations that involve increasing then decreasing patterns to make predictions
- graph increasing and decreasing patterns
- move between different representations (i.e., tables, graphs, expressions) of increasing and decreasing patterns
Sample Week at a Glance:
This sample week at a glance kicks off students’ formal introduction to increasing and decreasing patterns within Grade 6. Prior to this week, students would have come to the understanding that patterns involve repetition. What repeats can vary. For example, students will have answered open questions such as “Extend the pattern 5, 10, … in as many ways as you can.” Possible responses include:
- 5, 10, 5, 10, 5, 10, …
- 5, 10, 25, 5, 10, 25, 5, 10, 25, …
- 5, 10, 15, 20, 25, …
- 5, 10, 20, 40, 80, …
- 5, 10, 15, 25, 40, …
(Note that although all of the responses above are mathematically meaningful, students will focus on linear relations–response “c” above–in Grade 6.)
This open question ties together patterns in which elements themselves repeat and patterns in which operations repeat.
Provide students with pattern blocks or access to virtual manipulatives (e.g., Polypad by Mathigon, The Math Learning Center’s Pattern Shapes App). Tell a story in which tables and chairs are needed for a large party. Use a green triangle pattern block to represent one table. One person can be seated on each side of a triangular table. In turn, show that three people can sit at one table, six people can sit at two tables, and nine people can sit at three tables. Use a table of values to organize this information. Ask “How many people can sit at four tables? How do you know?” Have students turn-and-talk. Listen for strategies that involve addition (e.g., “nine and three more make twelve”) and multiplication (e.g., “four tables of three at each make twelve”). Select students to share these strategies. Annotate their thinking and add on to the table of values. Repeat for 5, 10, 20, 50, and 100 tables.
During. Tell students that tables come in different sizes: squares, trapezoids, and hexagons, which can be represented by orange, red, and yellow pattern blocks, respectively. (Note that a trapezoid can seat five people.) For each shape, have students build–then draw–the first three figures in the pattern. Ask “How many people can sit at 5, 10, 20, 50, and 100 of each table? How do you know?” Have students solve this problem in pairs. Encourage students to represent their ideas concretely (where practicable), pictorially, and symbolically.
Twenty people seated at five square tables
After. Make connections between repeated addition and multiplication. Highlight that to figure out the number of chairs needed for large numbers of tables, it is more efficient to multiply once than to add repeatedly. This strategy directly relates the two quantities, tables and chairs (or seated people). Return to the case of the triangular tables. Introduce the use of an expression, 3n, to describe the number of chairs needed for any number of triangular tables. Encourage students to write expressions to describe the number of chairs needed for the remaining shapes.
Before. Display the following patterns:
- 6, 11, 16, 21, 26, …
- 5, 10, 15, 20, 25, …
Ask “What is the same? What’s different?” Possible responses include:
- both patterns are increasing
- both patterns involve adding five each time
- the first pattern starts at six; the second pattern starts at five
- the second pattern contains multiples of five; the second does not
- the ones digit in both patterns repeats: 6, 1, 6, 1, 6, 1, … in the first and 5, 0, 5, 0, 5, 0, … in the second or even-odd in the first and odd-even in the second
- the tens digit in both patterns is both repeating and increasing: 0, 1, 1, 2, 2, 3, 3, …
Note that this warm-up foreshadows patterns that involve composite rules (i.e., 5n + 1). See the Instructional Routines page to learn more about Same But Different.
Return to yesterday’s tables and chairs problem. This time, tables can be joined together so that they share one side. As above, model the case of triangular tables, building to 102 people seated at 100 tables.
During. Provide students with pattern blocks or access to virtual manipulatives (e.g., Polypad by Mathigon, The Math Learning Center’s Pattern Shapes App). Invite students, in groups of three, to determine how many people can sit at 10, 20, 50, and 100 of one or all of the remaining shapes. Remind students to represent their ideas concretely (where practicable), pictorially, and symbolically. Ask “How does your pattern grow? Why does it grow in this way?” Monitor. Look for students making use of functional relationships. Encourage students to write an expression to describe each of their increasing patterns. Challenge them to explain the meaning, in context, of the coefficient and constant terms.
Twelve people seated at five square tables
After. Select students to share their strategies for determining the number of chairs needed for 100 square (202), trapezoidal (302), and hexagonal (402) tables. (You might decide to gradually build to 100 tables.) For example: “I chose hexagonal tables. Adding one table adds just four more seats since two sides are ‘lost’ when new and existing tables are joined. There are always two seats at the ends. So, 100 ⨉ 4 + 2 = 402.” Discuss the expressions for triangular (1n + 2), square (2n + 2), trapezoidal (3n + 2), and hexagonal (4n + 2) tables. Highlight that we can use these expressions to efficiently figure out how many chairs are needed for any number of tables.
Before. Display the following visual pattern:
Provide students with colour tiles or access to virtual manipulatives (e.g., Polypad by Mathigon, The Math Learning Center’s Whiteboard App). Have students build the first three figures in this pattern.
- How do you see the pattern growing?
- What comes next? Build it.
- What does Figure 10 look like? How many tiles? (3 ⨉ 10 = 30)
- What does Figure 100 look like? How many tiles? (3 ⨉ 100 = 300)
- What does Figure n look like? How many tiles? (3n)
- How many tiles in Figure 43? (3 ⨉ 43 = 129)
To learn more about Visual Patterns, see the Instructional Routines page.
During. Display the following visual pattern:
Note that this pattern involves a composite rule.
Invite students, in groups of three, to answer the set of questions above. Encourage students to represent their ideas concretely (where practicable), pictorially, and symbolically. Colour provides some support. If students are “stuck,” ask “What stays the same? (one red tile) What changes? (the number of blue tiles)” Again, look for students making use of functional relationships.
After. Select students to share their strategies. For example:
- I noticed that each figure has one red tile. I saw four ‘legs’ of blue tiles in each figure. The number of blue tiles in each leg matches the figure number. So, the expression is 4n + 1.
- I see four more tiles each time, one in each direction. Plus, there’s always one red tile. So, there are 43 ⨉ 4 + 1 = 173 tiles in Figure 43.
Ask “What comes before? What does Figure 0 look like?” (one red tile). Discuss.
Before. Display the following visual pattern:
Note that this pattern, again, involves a composite rule. Further, students might visualize how it grows in more complex ways–they might describe existing tiles shifting to make room for new tiles.
During. Have students, in groups of three, answer the following questions:
- How do you see the pattern growing?
- What comes next?
- What comes before?
- What does Figure 10 look like? Draw it. How many counters? (3 ⨉ 10 + 2 = 32)
- What does Figure 100 look like? Draw it. How many counters? (3 ⨉ 100 + 2 = 302)
- What does Figure n look like? How many counters? (3n + 2)
- How many counters in Figure 43? (3 ⨉ 43 + 2 = 132)
Encourage students to represent their ideas concretely (where practicable), pictorially, and symbolically. Challenge students to explain the meaning, in context, of the coefficient and constant terms: “How do you see 3? How do you see 2?”
After. Have students share their strategies. Draw out that the 3 and 2 in 3n + 2 can be seen as what changes and what stays the same, respectively. The expression also reveals the number of tiles in Figure 0.
What stays the same.
Before. Read Two of Everything by Lily Toy Hong. Introduce the concept of a function and the terms input and output. Like the Huktak’s magic pot, a function determines a particular output for a particular input. Revisit the part of the story when five coins are put into the pot and doubled. Create a table of values to record this information. Together, continue to add rows to the table of values using this doubling rule: “What if the Haktaks put 10 coins into the pot? 20? 50? 100? 43?” Introduce a new function (or “magic pot”), such as 5n or 2n + 3. Once again, build a table of values together.
During. Provide students with a table, like the one below. Have students, in groups of three, determine the rule for each “mystery function.” When students are ready to guess your rule, invite them to test their prediction by providing you with a new input. After providing students with the output–and if they are confident–have them guess your rule. Ask “How did you determine the rule?” Repeat with a different table of values. Challenge students to create their own “mystery function” for their partners if needed.
7n; 4n + 1; 3n + 2
After. Select students to share their strategies for determining the rule for each function. Students may use words (e.g., “the output is two more than triple the input”) or expressions (e.g., “3n + 2”). Highlight that the coefficient is related to the increase within the table (or rate of change) and that the constant is related to the output for an input of zero.
Next, students will explore increasing patterns in contextualized situations that, unlike the tables and chairs explorations above, cannot be modelled using manipulatives. For example, the relationship between cost and number of items or the relationship between savings and time. As above, students will describe patterns, predict what comes next and what comes ‘way down the line,’ and generalize functional relationships. Contextualized situations provide a way to introduce decreasing patterns. For example, students might explore the following: “Keira receives a $50 gift card to a local bubble tea shop. Each day, they order their favourite drink, which costs $6.” Students might then be asked to determine the balance of the card after 1, 2, 3, 8, or n visits.
Students will then be introduced to representing increasing and decreasing patterns using graphs. highlights relationships visually. Visual patterns and contextualized situations, such as those above, will be revisited. Students will make connections between graphs and other familiar representations (i.e., words, tables, expressions). Students will learn that graphs visually convey the mathematical ideas (e.g., what changes, what remains the same) within these other representations.
Suggestions for Assessment
By the end of Grade 6, students should be able to express increasing and decreasing patterns as functional relationships to make predictions. Students should be able to graph these relationships and explain how these various representations are connected.
Suggested Links and Resources
Key Number Concept 2: One-Step Equations
In Grade 6, students begin to use inverse operations (e.g., subtraction “undoes” addition) and the preservation of equality (i.e., subtracting three from both sides of an equation maintains “balance”) to solve equations. Students will continue to use this method in Grade 7 to solve two-step equations. Inverse operations will be helpful in later grades when students solve equations that involve three or more steps or that involve other types of numbers (e.g., integers, decimals, or fractions). Students will model and solve one-step equations using materials and diagrams such as algebra tiles, counters, balance scales, bar models, and so on; they will make connections between concrete, pictorial, and symbolic representations.
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of one-step equations in Grade 6:
- Understand that “=” means that two expressions are equivalent; it does not mean “the answer is…”
- Solve one-step equations using number sense and mental mathematics
- Represent unknowns using variables
- model and solve one-step equations using concrete materials
- develop an understanding of inverse operations and preservation of equality through concrete materials and pictorial diagrams
- transfer this understanding to symbolic representations (i.e., annotated equations)
Before. Ask the following open question:
Place the numbers from 1 to 9 in the boxes below to make the statement true. Each number can be used only once.
⬚ + ⬚ = ⬚ – ⬚
Note that there are multiple solutions (e.g., 1 + 3 = 6 – 2 or 3 + 5 = 9 – 1). Remind students that the equals sign means that the expressions on the left and right sides have the same values; it does not mean “the answer is.”
Tell students that they will be solving puzzles that challenge them to determine what is hidden in each diagram. Provide students with concrete materials such as cups and counters.
During. Display the following two puzzles:
x + 3 = 7
3x = 18
Have students work together In groups of three. Tell students that, when there is more than one cup, each covers the same number of paper clips (or counters). Monitor. Ask “How did you solve the puzzle?”and “How else might you solve the puzzle?” Provide students with additional pictorial puzzles as needed. Encourage students to show their thinking symbolically (i.e., using equations). Gradually transition from diagrams to equations as puzzles. Limit equations to those that involve addition and multiplication.. (Equations that involve subtraction and division will be introduced in the next lesson.) Include equations that involve numbers beyond known facts (e.g., x + 12 = 48 and 4x = 52) to nudge students toward using inverse operations.
After. Select students to share their strategies for determining what’s hidden in each puzzle. Students might use number sense and mental mathematics (e.g., “Four clips are hidden since 4 + 3 = 7”), act out inverse operations (e.g., “I took away three clips from both sides leaving the cup and four clips”), or annotate equations (e.g., x + 3 – 3 = 7 – 3 so x = 4). Note that rearranging objects on the same side preserves equality. Students can use this to visualize the inverse of multiplication (i.e., equal groups) as division (i.e., sharing):
3x = 18 so 1x = 6
Introduce symbolic notation that conveys preservation of equality:
Alternatively, algebra tiles can be used in place of cups and counters:
x + 3 = 7
3x = 18
Before. Display the following bar models or tape diagrams:
Ask” What equations do you see?” Note that the first shows both addition (i.e., 14 + 6 = 20) and subtraction (i.e., 20 – 6 = 14) and that the second shows both division (i.e., 20 ÷ 4 = 5) and multiplication (i.e., 4 ⨉ 5 = 20).
Once again, tell students that they will be solving puzzles that challenge them to determine unknown values in diagrams. This time, puzzles will be in the form of bar models or tape diagrams. Further, these pictures will represent equations that involve all four operations, not just addition and multiplication. Ask students to predict strategies to solve equations that involve subtraction and division (e.g., “Since subtraction ‘undoes’ addition, addition ‘undoes’ subtraction,” “To solve equations that involve division, we should use its inverse operation, multiplication.”)
During. Display the following:
Have students work together in groups of three. Ask “How did you solve the puzzle?”and “How else might you solve the puzzle?” Provide students with additional pictorial puzzles as needed. Encourage students to show their thinking symbolically (i.e., using equations). Gradually transition from diagrams to equations as puzzles. Include equations that involve numbers beyond known facts (e.g., x – 12 = 48 and x ÷ 4 = 13) to further nudge students toward using inverse operations.
After. Select students to share their strategies for determining unknown values in diagrams and equations. Emphasize the use of inverse operations. Highlight notation that communicates the preservation of equality. Show that mathematicians record the same operation on both sides to produce new equations.
Since equations are limited to one-step, which students have been working with beginning in Grade 3, this sequence of lessons may be sufficient for most students to achieve proficiency. Look for moments to embed variables and equations throughout the year. For example, students can use equations to model problems in contextualized situations. Introducing the instructional routine Numberless Word Problems can provide students with opportunities to practice using inverse operations and the preservation of equality.
Suggestions for Assessment
By the end of Grade 6, students should be able to model and solve one-step equations using inverse operations. Students should be able to represent the preservation of equality concretely, pictorially, and symbolically.