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 2
Patterns and Algebra
Students entering Grade 2 are expected to have an understanding of the language for labelling and describing repeating patterns. They are able to identify the pattern core of a repeating pattern, create their own patterns and extend patterns that they are given. They understand that knowing the pattern core can help them make predictions about how the pattern will continue and will be able to identify when there is or is not a pattern concretely. In Grade 2 students are making two big additions to their understanding of patterning and algebraic thinking. The first is the abstraction of pattern representation, from the concrete patterns of K and Grade 1 to the addition of pictorial representations in Grade 2. Secondly, students are moving from describing patterns and changes in quantity verbally, to also being able to reflect those patterns and changes numerically and pictorially. Although students are learning about symbolic representation in Grade 2, it is extremely important that these more abstract representations be firmly grounded in connection to concrete patterns, to keep student’s learning centred on a conceptual development of understanding how patterns work. Students need time to explore and develop understanding of important concepts, such as “all patterns repeat,” as they move to identifying and creating more complex repeating patterns and are introduced to the concept of growing patterns.
It is very important that students are exposed to many different types of patterns, so that they do not develop the incorrect idea that patterns all look the same. Another misconception that is common at this stage is that the equals symbol (=) means “the answer is…” Students need to be exposed to the symbolic representation expressed in a variety of ways in order to develop the correct meaning of balance. It is essential that students develop a strong understanding of the concrete, pictorial, symbolic connections in patterning, which they will continue to expand with the introduction of decreasing patterns in Grade 3, in order to have a strong conceptual foundation for the more abstract representations of patterns and algebraic thinking required in the intermediate grades. Games and routines, such as Guess My Rule, Visual Patterns, Choral Counting and Splat!, as well as lots of experiences creating patterns will help students build their understanding, while maintaining the concrete, pictorial, symbolic connections they require at this stage of development. This area of the curriculum is also a natural fit for integrating Indigenous topics and worldviews, as a huge variety of patterns can be investigated through art, nature, counting and language systems.
Key Concepts
Repeating and Increasing Patterns
Students build on their understanding of repeating patterns and extend that understanding to include increasing patterns, using a variety of formats including numbers, manipulatives, sounds and actions.
Change, Equality & Inequality
Students learn how to pictorially and symbolically represent a change in quantity using equations with missing numbers and tools such as ten frames. They will continue to deepen their understanding of equality, with particular attention to the = symbol.
Key Patterns and Algebra Concept 1: Repeating and Increasing Patterns
Overview
In Grade 2 students build on their understanding of repeating patterns to include finding the “core” in more complex patterns, such as circular and positional patterns. They extend that understanding to include increasing patterns. Students identify and extend patterns using a variety of formats including numbers, manipulatives, sounds and actions. As students move from describing the core of a repeating pattern to describing the consistent change of the steps/terms of a growing pattern, students are working towards the understanding that there is something about all patterns repeat. They build connections between the different representations of the same pattern and begin to understand that each representation shows us something different about the pattern and that understanding the pattern rule allows us to make predictions. Students need lots of practice identifying, creating and fixing patterns in many different formats to develop these essential understandings.
Note: Although we may used the term “pattern rule” in grade 2 informally, it is formally introduced in the Grade 3 BC curriculum standards. For grade 2, the focus is more on describing in words and numbers “what is happening” between terms in an increasing pattern.
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of patterns in Grade 2 and are the focus of learning in Kindergarten and Grade 1:
- Repeating patterns with multiple elements in concrete form
- “Reading” patterns by colour/shape/size
- Identifying the “core” of a repeating pattern
- Coding patterns orally using letter notation (ABBC)
- Translating patterns from one representation to another (red, blue, blue, green is ABBC is clap,snap, snap, stomp)
- Creating repeating patterns
- Extending repeating patterns by predicting what comes next
Progression:
- Read and describe complex repeating patterns (multiple attributes) that are presented visually/pictorially
- Notice and describe repeating patterns that are not presented in a line (circular, repeated within rows and columns eg: artwork, jewellery design)
- Identify and label the core of repeating patterns that have multiple elements or attributes in a complex presentation (within a design)
- Read, notice and describe that patterns can grow by repeating a consistent change, rather than repeat a core
- Identify and create increasing patterns using manipulatives, sounds, actions, and numbers (0 to 100)
- Translate and record (written, pictures) patterns in different forms using letter notation or other coding systems
- Predict what comes next, what comes before or what comes between (a missing part) in a repeating or growing pattern
- Discuss and compare patterns, sharing how two patterns are alike and how they are different, using mathematical vocabulary and language
Sample Week at a Glance:
Before the lessons in this week, students will have had experiences with a variety of complex repeating patterns (circular, positional, etc.) in context (artworks, buildings, etc.) They will have practiced identifying and describing these repeating patterns and have created their own. Games like Guess My Rule and routines like Visual Patterns will have been introduced using complex repeating patterns. There is an understanding that repetition is a key factor in determining if something is or is not a pattern. Students should also have practiced using story mats to create number stories and writing equations to reflect the story.
The purpose of this week is to introduce the concept of increasing patterns and the idea that there are other ways than a repeated “core” for that repetition to appear. This sample week would be placed towards the middle/end of a patterns unit.
Ask What do you know about patterns so far?
Read Flow, Spin, Grow: Looking for Patterns in Nature by Patchen Barss and Todd Stewart. Pause and ask the following questions
- Look: What patterns do you see?
- Breathe in deep: Is branching a pattern? What repeats?
- Grow: If something is growing what would make it a pattern? What repeats?
After reading the book, take students outside for a pattern hunt. Ask them to look for different types of patterns:
- How do they know it is a pattern? (What repeats?)
- Students can record their observations on clipboards or use tablets to take pictures.
- Teacher should interact with groups as they are doing their hunt. Ask: Why did you choose this? Why is it a pattern? How would you describe the pattern?
Gather students to consolidate what they learned: What new things did we learn about patterns today? What is the same about all patterns?
*Students could record what they learned in a journal or this could just be a discussion.
** Collect the images of a few growing patterns that the students collected for use in tomorrow’s lesson. Choose images that you think will help students clarify what counts as a growing pattern. It is beneficial to select 1 or 2 non-examples.
Project VIsual Patterns images for discussion.
- Lego Tower 1: What comes next? What would the 10th step/term be? Is this a pattern? How do you know? (This image is chosen to encourage students to discuss how the pattern is both repeating and growing.)
- Lego Tower 2: Ask the same questions as above. Students should notice that, while there is a possible red,green,red in the front, there is no pattern to the construction of the building.
Put students into groups of 3-4. Either project or hand out copies of the images that you chose from yesterday’s activity. Have groups sort the patterns into categories (repeating, growing, not a pattern).
After students have sorted the images, take the class on a gallery walk. * It is important that you have decided what you will highlight or ask at each stopping point. This can be done while circulating during the previous task. Focus on getting students to describe their reasoning for why something is or is not a pattern (Where is the repetition? Can they predict what comes next?)
- Skip count by 2, starting at 0. (5 columns) What patterns do you notice?
- Skip count by 2, starting at 1 (5 columns) What patterns are the same as when we start at 0? What patterns are different? What does this teach us about the “rule” for counting by 2s?
In groups of three, have students build and/or draw +2 patterns starting at different numbers. Have them label the steps and the number of items in each. Encourage them to think of equations that represent their pattern (0+2=2, 2+2=4).
- If students are understanding the equations, encourage them to find an equation that would describe the whole pattern ( 2+___ = ___)
Mathematicians use lots of different ways of representing patterns. Ask students to identify some of the ways they have represented patterns today (physical, pictures, numbers, equations).
How did you make an equation for the whole pattern? (choose students in advance based on their work/conversation during the group activity. Look for students that have found a way to represent the unknown.)
Read Two of Everything by Lily Toy Hong
- Either before or after reading, put students into groups of three.
- After reading, ask: What pattern rule does the pot follow? (doubling, twice as many out as in, etc.) Accept any logical answer. It is not important to focus on multiplication at this stage, but watch for students who mistakenly characterize the rule as add 2 or add 5.
In their groups of three, have students engage in thinking together:
- Ask: How many coins did the Haktaks start with? (5) How many did they have after Mr. Haktak threw the bag for the first time? (10)
- Give students coins or counters and ask them to build the first 2 steps of the pattern (remind them to leave space in between.
- Give the groups a number line to record the pattern on as well.
- Ask: What comes next? Wait while students discuss and build their answers.
- Ask: How could you figure out how much they would have if they threw the bag in 5 times? 5, 10, 20, 40, 80, 160
- Ask: What if the pot only added 2 each time?
*Questions can be asked of each group when they are ready to move on. If some groups finish quickly, they can come up with their own rules for the pot and build the patterns.
Mathematicians represent patterns in many ways.
Use student work that you select during collaboration time to show the same patterns represented on the number line, as a skip count, visually and (possibly) as equations.
Begin the lesson by reflecting on our learning so far this week.
- What have we learned about describing patterns this week?
- Record ideas on board or chart paper
In groups of two or three, have students take turns building patterns. The other partner(s) try to determine the pattern and come up with how to describe the pattern (some may refer to the pattern rule although this is not expected at Grade 2). They can then extend the pattern to check. *Encourage students to use different methods of building the patterns (numbers, objects, pictures, etc.)
*This is a good opportunity to support students, who are needing some direct instruction of the concept, but these students should not be pulled for the entire class, as they will also get great benefit from playing the game with their peers.
Closing Circle: Ask students to reflect upon what is important to consider when creating a pattern for others to solve or extend.
Any of these lessons may be stretched over two lesson periods, if students are actively exploring and need more time. It is essential not to skip consolidating the learning points following each activity.
As this week is designed to be positioned later in the unit/year, particular attention should be paid to formative assessment during the activities this week to determine if students need more practice with understanding what a growing pattern is and how to represent them. Since patterns are integral to mathematics, one approach at this stage is to move on to another topic such as computation, probability or geometry and include pattern activities within that learning. For example, grade 2 students learn about days, weeks, months and years and how to use a calendar. There is lots of potential for pattern explorations in this topic. In geometry, grade 2s learn about the properties of 2D & 3D shapes and could look at patterns of sides, faces, etc. Patterns are also an important part of data and probability.
Suggestions for Assessment
Look for evidence learning throughout the week. Evidence can be anecdotal observations, elicited by interview or targeted questioning, products and exit slips. Consider where the student is at the end of the week and what their next steps will be. Students who are still struggling to build a growing pattern, will need to practice over the next few weeks and additional evidence of learning collected. This can be done by building patterns for 10 minutes a day with an adult or a peer, until they become comfortable with the concepts.
By the end of Grade 2, students will be able to:
- Create, describe, label and extend complex repeating patterns
- Represent growing patterns in different forms: build, draw, numbers, nature
- Connect growing patterns to each other
- Develop, demonstrate, and apply pattern concepts through play and inquiry
- Identifies increasing patterns using manipulatives, sounds, actions, and numbers (0 to 100)
Suggested Links and Resources
Flow, Spin, Grow: Looking for patterns in Nature by Patchen Barss and Todd Stewart
Visual Patterns (how to and resources): https://mathingaround.com/resources/educator-resources/routines/visual-patterns/
Two of Everything by Lily Toy Hong
Key Patterns and Algebra Concept 2: Change, Equality & Inequality
Overview
Building on their ability to describe the change in quantity verbally using concrete materials in Grade 1, Grade 2 students learn how to pictorially and symbolically represent the change using equations, missing numbers (variables), ten frames, hundred charts and other appropriate templates. At this stage students can represent a concrete change such as, “if I build 10 and add 4 more, I will have 14,” as 10+__=14 or 14=10+__. They will also be able to represent changes in quantity to numbers beyond 20, up to 100.
Students in Grade 2 continue to deepen their understanding of equality, with particular attention to the = symbol. As they begin to represent relationships using equations, it is important that they learn that = means balance and does not always come at the end of an equation. Students in Grade 2 may have developed the incorrect idea that the equals symbol means “the answer is…” and need to see it used in a variety of ways to help them develop a true understanding of equality.
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of change, equality, and inequality in Grade 2 and are developed in Kindergarten and Grade 1:
- Change in quantity to 20 using concrete materials and verbal descriptions
- Composing and decomposing quantities to 20
- Recording equations symbolically using = and ≠ to describe
Progression:
- Visually represent changes in quantity using ten frames, hundreds charts, etc.
- Records changes in quantity numerically (7+__=18)
- Uses symbolic representation to express equality (=)
- Understands that the equals sign (=) means balance and can occur at any place in a number sentence as long as each side is actually equal. For example, students learn that 10=2+5+3 and 10=2+5+3=7+3=10 are correct, but 10=2+5=7+3=10 is not.
Sample Week at a Glance
This week could follow the sample week from Repeating and Increasing Patterns or be taught at another time. In either case, students need to be familiar with the concept of addition and with the symbolic notation of writing addition equations (2+3=5). They should have previous experience with creating equations to go with number stories and with creating their own stories and equations.
Introduce the balance scale: https://mathigon.org/polypad/33MaM15KgWqpKw
Project the activity or use a physical balance scale and numbered weights.
Put the largest number on the right hand side of the scale (5). Ask: How can we balance the scale? Once students have added the equivalent of 5 to the other side, ask: how could we write this as an equation? *If students swap the numbers around so that the single number is after the = sign ask: Is that how it is represented on the scale? Say: Equal means balance.
In groups of 3 have students play with the polypad activity (or with physical balances). Encourage them to build creatively and record equations where the equals sign is in different places. Circulate during this time to watch for misconceptions and to gather interesting examples for the consolidation. Once groups can build equal equations in different formats, ask them to do some for inequality using the ≠ symbol in their equations.
To consolidate student learning, discuss two or three of the student created examples. Give students only the picture or only the equation and ask them to provide the other. If there are no examples of 3=3, 3+2+5=10 or other alternatives to ___+___=____, you can provide them.
Ask: What does = mean? What does ≠ mean?
*The answer should be about balance, rather than about an answer.
Number Talk: 20=____
Record different solutions that students offer and ask them to explain how they determine equality.
Open Middle Problem: Give groups of three students the printed template and counters to work with. If they need a greater challenge, ask them how they would make the template harder to complete (more boxes/equal signs, bigger numbers) and have them work on that.
Exit slip/math journal: Use pictures, numbers and words to explain the meanings of equality and inequality.
Read Pigeon Math by Asia Citro
Stop at a few points in the story to ask students to write an equation that represents what is happening on the page. Stop at one or two places before the pigeons arrive or fly away, tell students that “some” arrive or fly away (without giving the actual number) and how many are there after that happens and ask students to write an equation to represent their prediction such as 9+__=15.
In groups of two, have students use story materials to create their own number story contexts. This part of the lesson can be integrated into Language Arts if you wish to also have students complete the Story Workshop writing and sharing process. Have students create three equations with missing numbers and three action descriptions that can be represented on their mats. Groups can then switch mats and questions with another group and solve/represent the scenarios suggested.
*Circulate to record observations and select students to share helpful strategies during consolidation.
Invite students to share what math knowledge and strategies helped them figure out the solutions. Discuss as a whole class and then have students complete individual exit slips/journals.
Explain to students that mathematicians use patterns to help them understand the rules of math. These patterns are called “generalizations.” Generalizations are things that we know will always be true when we are doing math.
Ask: What are some generalizations (things you know will always be true) about adding that help us find the missing numbers in equations? (each side of the equation must be the same value, + means that we are putting the numbers together). Watch for misconceptions like, “the answer is always bigger.” Ask: Where is the answer? What do you mean by “bigger”? Can anyone think of a time where the answer might NOT be bigger?
Have the same conversation about subtraction.
In groups of three have students explore the following statements one at a time for about five minutes each. Following each exploration have a quick discussion to determine if the statement is always, sometimes or never true.
- When you add or subtract 2 numbers, it does not matter which number you start with.
- When you add or subtract 0 from a number, it does not change the value.
- When I add more than 2 numbers together, I can add them in any order.
Exit Slip: What generalizations help me add and subtract?
Since the understandings in this week are fairly straightforward and build on previous content from Grade 1, students will likely have a proficient understanding at this point. It is important, however, to expose students to equations and number stories throughout the year, so that they can develop intuition and fluency through working with numbers and equations in a variety of contexts.
Suggestions for Assessment
Watch and listen for students who are understanding the nature of equality and the basic tenets of adding, but are not yet able to subtract or students who add and subtract visually, but are not yet comfortable with the symbolic representation (equations). These understandings are elements of computational fluency, as well as patters, and should be revisited through the use of routines like Number Talks and problem solving activities throughout the year.
By the end of Grade 2, students will be able to:
- Uses pictures, numbers and words to explain equality as a balance
- Uses the symbols = and ≠ correctly, in a variety of equation formats (n=n, n+N=a, a=N–n, etc.)
- Shows changes in quantity numerically (addition/subtraction equations with missing numbers)
Suggested Links and Resources
- Mathigon Polypad balance scale HERE
- Open Middle Problem HERE
- Pigeon Math by Asia Citro
- Always, Sometimes, Never statements HERE
- Mathigon Activities https://mathigon.org/task/equality-with-number-bars
- SPLAT! https://stevewyborney.com/2017/02/splat/