### 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.

## Patterns and Algebra

Students are increasing their fluency with pattern rules and one-step equations from previous years. They go from describing pattern rules in repeating patterns using “the core” (e.g., AB pattern) in K through grade 2, to describing increasing patterns in grade 2, and then layering in decreasing patterns in grade 3. These patterns come in many forms: number lists, visual patterns, sounds, concrete materials, charts, graphs, tables, etc. In grade 5, students use an equation with a variable to represent the unknown quantity as one way to describe the pattern rule. They should still be able to describe the rule using words or to represent it in a table, too. In this context, the “variable” being used to express what is happening in the pattern is building on what the previous term, step or figure is. So an “add 2” pattern might be represented as p+2 to represent previous term plus two more.

Meanwhile, students are building on their fluency with solving one-step equations from Grades 3 and 4. These are equations of various forms: start unknown (e.g., n + 15 = 20), change unknown (e.g., 12 + n = 20), result unknown (e.g., 6 + 13 = n). They include all operations in Grades 4 and 5. This year, there is an emphasis on using a variable to represent an unknown number in grade 5 (e.g., 5n = 35 rather than 5 × __ = 35). Students go further this year by expressing a context or problem as an equation using symbols (e.g., n + 4 = 15). That is, students may find the solution to a problem by writing and solving an algebraic equation.

Students continue to use algebraic expressions to describe pattern rules in grade 6 and beyond but include a functional use of variables such as 3x or 4n-2. They continue to solve one-step equations in Grade 6, too, before moving into two-step equations in Grades 7 and 8. The emphasis in Grades 6 -8 is on solving equations using the preservation of equality (e.g., “What you do to one side of the equation, you do to the other side.”)

### Key Concepts

#### Pattern Rules

Students build on their fluency with describing rules for increasing and decreasing patterns with words (e.g., Start at 5 and add 2 each time) from grade 4. They go further this year by using numbers, symbols, and variables to represent pattern rules.

#### One-Step Equations

Students become more fluent with solving one-step equations which they began in Grades 3 and 4. There is an emphasis on using a variable to represent an unknown number (e.g., 5n = 35) in this grade. Students go further this year by expressing a given problem as an equation using symbols (e.g., n + 4 = 15).

#### Key Patterns and Algebra Concept 1: Pattern Rules

##### Overview

Students should already be familiar with describing repeating patterns using the core, as well as describing increasing and decreasing patterns from previous years. Doing this in a variety of ways is important. For example, students should be able to represent a visual pattern (see below) using numbers (i.e., the number of squares in each “term”) in a chart, can draw or describe what comes next and how they know, and be able to extend the pattern, too. We want students to be able to describe the pattern rule in words (e.g., “Start at 5. Add 4 each time.”) as well as beginning to explore algebraic representations..

Often, there is more than one pattern evident. Students usually notice the recursive pattern first, which allows students to predict what comes next based on the previous number or visual. For example, they may say “add 4 each time”, which makes predicting the next number easier. However, predicting the 100th term is more time consuming with the recursive pattern. Some students may begin to explore and investigate the functional relationship within an increasing or decreasing pattern (e.g., “Multiply the term by 4, then add 1”) and this will help students to predict any number. This is not formally expected at the Grade 5 level in our BC curriculum standards.

##### Patterns and Algebra Foundations:

The following concepts and competencies are foundational in supporting understanding of pattern rules in grade 5:

• Skip counting forwards and backwards by a variety of numbers using any starting point
• Addition and Subtraction facts to 20
• Multiplication and Division facts to 100
• Represent patterns in concrete, pictorial, and symbolic forms
• Connecting number patterns to algebraic equations and to solving equations
##### Progression:
• Exposure to a variety of patterns (repeating, increasing, decreasing) in different forms (words, numbers, visuals, sounds, movement, tables, charts, graphs, etc.)
• Describe the pattern (e.g., “How do you see it changing?”)
• What comes next?
• Focus on recursive patterns (while continuing the above)
• Explore functional relationships in patterns (not expected at Grade 5)
• Representing patterns in different forms (while continuing the above)
• Visual to Table
• Table to Equation
##### Sample Week at a Glance:

Prior to this week, students will have been building fluency with their math facts through such instructional routines as: and Problem Strings. Skip counting routines are also useful when exploring recursive patterns. Instructional routines such as Choral Counting. This routine can be extended to find patterns and is referenced below, too.

Before: Visual Patterns with this image. How do you see the pattern changing? Can you draw what comes next? How many squares in the next image? in the 10th image? in the 100th image? (challenge)

During: Students are asked to determine how many handshakes there would be if every student in the class shakes hands with each other student exactly once. Students can act it out, draw pictures, use a chart, etc. to solve the problem. They can do this alone, in pairs, or small groups.

Adaptations include asking students to solve the problem with a smaller number of students or determining the answer in a systematic way for 2 students, 3 students, 4 students, …. Then look for a pattern and generalize.

Extensions include asking students to solve the problem for any number of students and to describe the pattern rule for finding the answer to this problem.

After: Teacher facilitates a class discussion where students share their strategies, findings, and solutions with one another. How is this problem related to the Visual Pattern activity we did at the beginning of class?

Lesson Topic: Exploring Patterns

Before: A pattern begins with 2, 4, …. How might it continue? How many different ways can you do this? What kind of pattern is each? What is the pattern rule for each? Students do this as a think-pair-share.

During: Math Workshop. Some ideas include:

• Patterns with materials (e.g., pattern blocks)–create and describe your own patterns
• Number patterns worksheet (increasing & decreasing) asking students to write the next few terms in the sequence
• Sound patterns (listening to music, clapping, instruments, reading music sheets)
• Movement patterns using technology (e.g., lightbot as one example)

Alternatively, the teacher asks students to create as many different types of patterns as they can using a variety of formats: numbers, letters, pictures, sounds, motion, etc. Teacher provides materials. Students do a gallery walk to get new ideas.

How can we classify our patterns? (repeating, increasing, decreasing) Pick one or more of your patterns. How can you describe the pattern rule? (words vs algebra). Students can work in pairs or alone.

The openness of this task makes it naturally differentiated. You may need to help students who struggle with open tasks to get started by providing the beginning of a pattern that they continue.

After: Teacher facilitates a class discussion where some students share a pattern and how they determined the pattern rule.

Lesson Topic: Recursive Patterns

Before: For example, skip count by 3s, starting at 5, and write numbers vertically in columns of five. How can you use the patterns in the chart to predict future numbers? Use this to introduce the idea of a recursive pattern.

During: Provide students with several tables (on the whiteboard or screen) with simple increasing and decreasing patterns (3 to 5 examples is good). Ask students to find the next three numbers in each table. Students share their strategy. Focus on the recursive pattern first (how students used the previous number to predict the next one). Most students will have done it this way.

You could then extend the experience to determine the functional relationship/s within the pattern; that is, how knowing the term number can predict the unknown number. Students can work in pairs. Further extend this experience by having students determine an algebraic expression.

After: Students share their findings as a class. Teacher highlights the connection between the recursive pattern (e.g., add 2 to number each time) and functional pattern (term × 2 + 3 = number) in each example. See image below for an example. Showing this connection is a key understanding for helping students to predict functional patterns more easily but is not required or expected in our Grade 5 BC curriculum but will be formally developed in Grade 6.

Lesson Topic: Pattern Rules

Before: Guess My Rule instructional routine from High Yield Routines (NCTM). You may provide a few examples. Connect this to yesterday’s lesson about recursive patterns.

During: Math Workshop. Some ideas include:

• Finding the Pattern Rule recording form
• Meeting with the teacher to review understanding of pattern rules
• Online practice with pattern rules (e.g. Mathletics or other online platform)
• Problem-solving task or describe a context for a given pattern rule
• Throwback Thursday activity to review previously taught concepts

After: Class share on Math Workshop. Then give students an exit ticket to see if they are able to find the pattern rule and explain how they did it. Collect these for formative assessment.

Lesson Topic: Representing Patterns

Before: Based on the exit ticket from yesterday, today’s lesson may need further review of yesterday’s lesson to start. Otherwise, choose a pattern to display from Visual Patterns, such as this. How can we represent this information in a table? How can we predict the number of cubes in the next image? 10th figure? 100th figure? nth figure? Discuss strategies as a class.

During: Provide students with other visual patterns* and have them work in partners to answer the same questions above. Provide materials for students who would like to build their patterns. Extend students by asking them to create their own pattern that matches an equation that you give them (e.g., number of cubes = 3n – 1).

*Carefully select visual patterns so that the equation that describes the pattern is linear (i.e., y = mx + b). Many examples on this site are NOT linear patterns.

After: As a class, have students share and discuss solutions. Provide an exit ticket or have students record a consolidating example in their math journals as formative feedback.

Based on your formative feedback from Friday’s lesson, you may find that students would benefit from further practice or perhaps some small group instruction and assessment tasks. Otherwise, the next week would move into solving one-step equations, which is our next key concept.

##### Suggestions for Assessment

What to look for:

• Can show or describe what comes next in a pattern
• Can describe the pattern rule to a pattern they created
• Can determine the recursive pattern

By the end of Grade 5 students will be able to:

• Describe the pattern rule using words, numbers, symbols, and variables
• Describe and interpret visual or concrete increasing and decreasing patterns, generalizing what the pattern rule is (ie. +5 or x2)

#### Key Patterns and Algebra Concept 2: One-Step Equations

##### Overview

Students have been solving one-step equations since Grade 3 with addition and subtraction, and again in Grade 4 including multiplication and division, too. In Grade 5, we emphasize solving one-step equations with variables (as opposed to a “blank” or other symbol). Again, the variable is in various forms: start unknown (e.g., n + 15 = 20), change unknown (e.g., 12 + n = 20), result unknown (e.g., 6 + 13 = n). You may introduce the notation 3n to mean 3 × n. The reason for this is that 3n is quite literally 3 n’s, and combining 3 n’s means n + n + n which equate to 3 × n. Be sure to emphasize that the “=” sign means “has the same value or quantity as” and not “the answer is” for which many students believe. Writing equations in different formats, such as 4 + 5 = 9 as 9 = 4 + 5 helps students to make sense of this, since “the answer is” does not make sense for the second equation.

At Grade 5, it is expected that students will solve equations “by inspection” and use number sense and mental math strategies to solve one-step equations.  In Grade 6, they will be introduced to more sophisticated strategies such as using inverse relationships and preserving equality.

Students extend their understanding of one-step equations by using them to model a context or problem. For example, they can solve a problem by first writing an equation (e.g., n + 4 = 15) and then solve the equation to answer the problem. Students should also be able to write a story problem that can be solved by a given one-step equation.

##### Patterns and Algebra Foundations:

The following concepts and competencies are foundational in supporting understanding of one-step equations in Grade 5:

• The meaning of equality
• Representing an unknown quantity with a variable
• Addition and Subtraction facts to 20
• Multiplication and Division facts to 100
• Understanding the various meanings of the different operations (e.g., subtraction as take-away or comparison or missing addend). Marian Small’s book is a good reference for this (see Resources below)
• Connecting number patterns to algebraic equations and to solving equations
##### Progression:
• Review of foundational algebraic ideas such as equality and variable
• Concrete or visual models for one-step equations, such as balance scale or bar models, involving addition and subtraction
• Connecting algebra to these models
• Solving one-step equations without the need for concrete or visual models
• Solving one-step equations using all four operations
• Answer word problems by solving one-step equations
• Providing a context for one-step equations
##### Sample Week at a Glance

Prior to this week, students will have reviewed the foundational algebraic ideas outlined in the learning progression above. They will have already studied pattern rules for increasing and decreasing patterns.

Lesson Topic: Introduction to Bar Models

Before: Bar model puzzles. Show students the images below and ask them to determine the lengths of the unknown bars. How do they know?

Adapt by providing concrete bars like Cuisenaire rods or linking cubes. Extend by asking students to write an equation that connects to each bar model. Discuss these ideas as a class: How students determined the missing lengths and how the bar models match their algebraic equations.

During: Provide students with additional bar model puzzles as well as their corresponding equations, mixed up. Ask them to match the bar model to the correct equations and explain why they did this. You may provide additional equations to extend student thinking further. For example, the leftmost bar model above can be represented by 2 + n = 9 as well as 9 – n = 2. Alternatively, ask students who are done early to come up with a different equation that matches each bar model.

After: Facilitate a class discussion on how students knew which equation(s) matched which bar model. Have students compare and contrast the different bar models and matching equations to see how the numbers, operations, and variables in the equations correspond to each part of the corresponding bar model.

Extend students’ thinking by asking them to draw their own bar model for a given equation (e.g., n + 4 = 10) or expression (3 + n). Some challenge questions could involve multiplication and division equations, which we will explore later.

Lesson Topic: Solving Addition & Subtraction Equations

Before:  Give students an equation such as n + 8 = 15. Ask how they would represent this with a bar model. Ask students to share and explain other strategies they have for solving this question.

During: Math Workshop. Some ideas include:

• Meeting with students in a small group to check in and provide support with solving one-step equations with bar models and mental math strategies
• A solving one-step equations worksheet (just addition and subtraction)
• Online practice such as solving one-step equations
• Bar model puzzles (addition and subtraction)
• Addition and subtraction word problems, such as from Tang Math word problems generator

After: Meet as a class to discuss Math Workshop. Have students do an exit ticket or write an entry in their math journals showing how to solve a one-step addition or subtraction question using bar models and algebraic steps.

Lesson Topic: Solving Multiplication & Division Equations

Before: How would you represent 2n = 6 using a bar model? Have students provide their conjectures, compare with other students, and critique as a class. Key ideas include that 2n mean two n’s or n + n which equals 2 × n. How does this connect to the image below?

During: Have students write bar models for several other equations to check understanding, such as 5n = 10. Challenge: How would you model n/5 = 10? We want students to use bar models as a tool for understanding algebra rather than another thing to memorize. This is why it’s a good idea to let students try to come up with the bar model themselves.

Adapt by asking students to use bar models to represent expressions to start: n, n + 3, 2n, 2n + 1, etc. before modeling equations such as n = 5, n + 3 = 8, 2n = 10, 2n + 1 = 15 (challenge). Extend by asking students to model division equations or two-step equations.

After: Provide students with the opportunity to practice solving one-step equations using algebraic steps as well as bar models, such as on a provided recording form or questions..

Lesson Topic: Solving One-Step Equations (All Operations)

Before: Number Talk. Ask students to do one or both options. Option 1: 15 × 6 or Option 2: 90 ÷ 6. Discuss strategies. How are these two options related? Emphasize the idea of opposite operations and significance here.

During: Math Workshop. Some ideas include:

• Meeting with students in a small group to check in and provide support with solving one-step equations with bar models or using mental math strategies
• A solving one-step equations worksheet (all operations)
• Online practice such as solving one-step equations
• Bar model puzzles (all operations)
• Mixed word problems, such as from Tang Math word problems generator

After: Meet as a class to discuss Math Workshop. Have students do an exit ticket or write an entry in their math journals showing how to solve a one-step multiplication question using bar models and algebraic steps.

Lesson Topic: Solving Word Problems with One-Step Equations

Before: I Have, You Need. One way to use instructional routines is as a way of reviewing previously learned concepts. They do not have to be directly linked to the day’s lesson.

During: Provide students with two or three word problems such as those found here. Ask students to write an equation that matches the context of each problem. How do you know your equation is suitable? Students often struggle with which operation to choose because they do not have a clear understanding of the different meanings of each operation. You may need to review these (Marian Small’s book is a good reference).

After: Provide students with mixed problems such as those from here to practice writing and solving equations to answer word problems. As an exit ticket, provide an equation, such as n + 5 = 8 or 4n = 12, and ask students to write a word problem that can be answered by solving the given equation.

Based on Thursday and Friday’s exit tickets or journal entries, you may find students need more practice solving one-step equations or with word problems. You can also do a lesson connecting the two key concepts of patterns and algebra. For example, use a simple visual pattern such as the one below and have students write and solve a one-step equation to determine the number of squares in the 100th term.

##### Suggestions for Assessment

What to look for:

• Represents an equation using a bar model and vice versa
• Can solve an equation by inspection (using number sense and mental math strategies)
• Can solve an equation using a bar model
• Can solve an equation using algebraic steps
• Connects the algebraic steps of solving an equation with the bar model
• Can solve word problems using an equation
• Can write a word problem given an equation

By the end of Grade 5 students will be able to:

• Model equations using bar models and vice versa
• Connect the parts of a bar model with the corresponding parts of an equation
• Solve equations by inspection using number sense and mental math, using bar models, and with algebraic steps
• Solve word problems by solving an equation